Question 12 Marks
Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\text{x}+\Big(\frac{\text{dy}}{\text{dx}}\Big)=\sqrt{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2}$
Answer$\text{x}+\Big(\frac{\text{dy}}{\text{dx}}\Big)=\sqrt{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2}\Rightarrow\Big(\frac{\text{dy}}{\text{dx}}\Big)=\Big(1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2\Big)^{\frac{1}{2}}$Squaring both sides, we get
$\Rightarrow\Big(\text{x}+\frac{\text{dy}}{\text{dx}}\Big)^2=1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2$
$\Rightarrow\text{x}^2+2\text{x}\frac{\text{dy}}{\text{dx}}+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2=1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2$
$\Rightarrow2\text{x}\frac{\text{dy}}{\text{dx}}+\text{x}^2=1$
In this differential equation, the order of the highest order derivative is 1 and the power is 1. So, it is a differential equation of order 1 and degree 1.
Hence, it is a linear differential equation.
View full question & answer→Question 22 Marks
Form the differential equation from the following primitives where constants are arbitrart:$\text{y}^2=4\text{ax}$
AnswerThe equation of family of curves is $\text{y}^2=4\text{ax}$ where a is an arbitrary constant. This equation contains only one arbitrary constant, so we shall get a differential equation of first order. Differentiating equation (1) with respect to x, we get $2\text{y}\frac{\text{dy}}{\text{dx}}=4\text{a}$ $\Rightarrow\frac{\text{y}}{2}\frac{\text{dy}}{\text{dx}}=\text{a}\ ...(2)$ Putting the value of a in equation (1), we get $\text{y}^2=4\frac{\text{y}}{2}\frac{\text{dy}}{\text{dx}}\text{x}$ $\Rightarrow\text{y}=2\text{x}\frac{\text{dy}}{\text{dx}}$It is the required differential equation.
View full question & answer→Question 32 Marks
Form the differential equation from the following primitives where constants are arbitrart:$\text{y}=\text{ax}^2+\text{bx}+\text{c}$
AnswerThe equation of family of curves is
$\text{y}=\text{ax}^2+\text{bx}+\text{c}\ ...(1)$
where a, b and c is an arbitrary constant. so, we shall get a differential equation of third order.
Differentiating equation (1) with respect to x, we get
$\frac{\text{dy}}{\text{dx}}=2\text{ax}+\text{b}\ ...(2)$
Differentiating equation (2) with respect to x, we get
$\frac{\text{d}^2\text{y}}{\text{dx}}=2\text{a}\ ...(3)$
Differentiating equation (3) with respect to x, we get
$\frac{\text{d}^3\text{y}}{\text{dx}3}=0$
It is the required differential equation.
View full question & answer→Question 42 Marks
Form the differential equation from the following primitives where constants are arbitrart:$\text{xy}=\text{a}^2$
AnswerThe equation of family of curves is
$\text{xy}=\text{a}^2\ ...(1)$
where a is an arbitrary constant.
This equation contains only one arbitrary constant, so we shall get a differential equation of first order.
Differentiating equation (1) with respect to x, we get
$\text{y}+\text{x}\frac{\text{dy}}{\text{dx}}=0$
It is the required differential equation.
View full question & answer→Question 52 Marks
Form the differential equation from the following primitives where constants are arbitrart:$\text{y}=\text{cx}+2\text{c}^2+\text{c}$
AnswerThe equation of family of curves is $\text{y}=\text{cx}+2\text{c}^2+\text{c}\ ...(1)$ where c is an arbitrary constant. This equation contains only one arbitrary constant, so we shall get a differential equation of first order. Differentiating equation (1) with respect to x, we get $\frac{\text{dy}}{\text{dx}}=\text{c}\ ...(2)$ $\Rightarrow\frac{\text{y}}{2}\frac{\text{dy}}{\text{dx}}=\text{a}\ ...(2)$ Putting the value of a in equation (1), we get $\text{y}=\text{x}\frac{\text{dy}}{\text{dx}}+2\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+\Big(\frac{\text{dy}}{\text{dx}}\Big)^3$It is the required differential equation.
View full question & answer→Question 62 Marks
Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=(\cos^2\text{x}-\sin^2\text{x})\cos^2\text{y}$
Answer$\frac{\text{dy}}{\text{dx}}=(\cos^2\text{x}-\sin^2\text{x})\cos^2\text{y}$
$\frac{\text{dy}}{\cos^2\text{y}}=(\cos^2\text{x}-\sin^2\text{x})\text{dx}$
$\int\sec^2\text{y dy}=\int\cos2\text{x dx}$
$\tan\text{y}=\frac{\sin2\text{x}}{2}+\text{C}$
View full question & answer→Question 72 Marks
Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\frac{\text{d}^4\text{y}}{\text{dx}^4}=\Big\{\text{c}+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2\Big\}^{\frac{3}{2}}$
Answer$\frac{\text{d}^4\text{y}}{\text{dx}^4}=\Big[\text{c}+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2\Big]^{\frac{3}{2}}$
$\Rightarrow\Big(\frac{\text{d}^4\text{y}}{\text{dx}^4}\Big)^2=\Big[\text{c}+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2\Big]^3$
$\Rightarrow\Big(\frac{\text{d}^2\text{y}}{\text{dx}^4}\Big)^2=\text{c}^3+\Big(\frac{\text{dy}}{\text{dx}}\Big)^6+3\text{c}\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+3\text{c}^2\Big(\frac{\text{dy}}{\text{dx}}\Big)$
$\Rightarrow\Big(\frac{\text{d}^2\text{y}}{\text{dx}^4}\Big)^2-\Big(\frac{\text{dy}}{\text{dx}}\Big)^6-3\text{c}\Big(\frac{\text{dy}}{\text{dx}}\Big)^2-3\text{c}^2\Big(\frac{\text{dy}}{\text{dx}}\Big)-\text{c}^3=0$
The highest order differential coefficient is $\Big(\frac{\text{d}^2\text{y}}{\text{dx}^4}\Big)$ and its power is 2.
It is a non-linear differential equation with order 4 and degree 2.
View full question & answer→Question 82 Marks
Find the equation of the curve which passes through the point (3, -4) and has slope $\frac{2\text{y}}{\text{x}}$ at any point (x, y) on it.
AnswerAccording to the quation,
$\frac{\text{dy}}{\text{dx}}=\frac{2\text{y}}{\text{x}}$
$\Rightarrow \frac{1}{2\text{y}}\text{dy}=\frac{1}{\text{x}}\text{dx}$
Integrating both sides with respect to x, we get
$\int\frac{1}{2\text{y}}\text{dy}=\int\frac{1}{\text{x}}\text{dx}$
$\Rightarrow \frac{1}{2}\log|\text{y}|=\log|\text{x}|+\text{C}$
Since the curve passes though (3, -4), it satisfies the above equation.
$\therefore \frac{1}{2}\log|-4|=\log|\text{x}|+\text{C}$
$\Rightarrow \log |2|-\log|3|+\text{C}$
$\Rightarrow \text{C}=\log|\frac{2}{3}|$
Putting the value of C, we get
$\log|\text{y}|=2\log|\text{x}|+2\log|\frac{2}{3}|$
$\log|\text{y}|=\log|\frac{4}{9}\text{x}^{2}|$
$\text{y}=\pm\frac{4}{9}\text{x}^{2}$
$9\text{y}-4\text{x}^{2}=0$
The given point does not satisfy the equation $9\text{y}-4\text{x}^{2}=0$
$\therefore 9\text{y}+4\text{x}^{2}=0$
View full question & answer→Question 92 Marks
Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+\frac{1}{\frac{\text{dy}}{\text{dx}}}=2$
Answer$\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+\frac{1}{\Big(\frac{\text{dy}}{\text{dx}}\Big)}=2$
$\Rightarrow\Big(\frac{\text{dy}}{\text{dx}}\Big)^3+1=2\Big(\frac{\text{dy}}{\text{dx}}\Big)$
$\Big(\frac{\text{dy}}{\text{dx}}\Big)^3-2\Big(\frac{\text{dy}}{\text{dx}}\Big)+1=0$
This is a polynomial in $\frac{\text{dy}}{\text{dx}}.$
The highest order differential coefficient is $\frac{\text{dy}}{\text{dx}}$ and its power is 3.
So, it is a non linear differential equation with order 1 and degree 3.
View full question & answer→Question 102 Marks
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}+\text{y}}+\text{x}^2\text{e}^\text{y}$
Answer$\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}+\text{y}}+\text{x}^2\text{e}^\text{y}$
$=\text{e}^\text{x}\text{e}^\text{y}+\text{x}^2\text{e}^\text{y}$
$\frac{\text{dy}}{\text{dx}}=\text{e}^\text{y}(\text{e}^\text{x}+\text{x}^2)$
$\int\text{e}^{-\text{y}}\text{dy}=\int(\text{e}^\text{x}+\text{x}^2)\text{dx}$
$-\text{e}^{_\text{y}}=\text{e}^\text{x}+\frac{\text{x}^3}{3}+\text{C}$
View full question & answer→Question 112 Marks
If sinx is an integrating factor of the differential equation $\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$ than write the value of P.
AnswerIt is given that sinx is the intergrating factor of the differential equation $\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}.$
$\text{e}^{\int\text{P}\text{dy}}=\sin\text{x}$
$\Rightarrow \int \text{P}\ \text{dx}=\log|\sin\text{x}|$
$\Rightarrow \int \text{P}\ \text{dx}=\int\cot\text{x}\ \text{dx}$
$\Rightarrow \text{P}=\cot\text{x}$
View full question & answer→Question 122 Marks
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}+2\text{x}=\text{e}^{3\text{x}}$
Answer$\frac{\text{dy}}{\text{dx}}+2\text{x}=\text{e}^{3\text{x}}$
$\frac{\text{dy}}{\text{dx}}=\text{e}^{3\text{x}}-2\text{x}$
$\int\text{dy}=\int(\text{e}^{3\text{x}}-2\text{x})\text{dx}$
$\text{y}=\frac{\text{e}^{3\text{x}}}{3}-\frac{2\text{x}^2}{2}+\text{c}$
$\text{y}=\frac{\text{e}^{3\text{x}}}{3}-\text{x}^2+\text{c}$
$\text{y}+\text{x}^2=\frac{1}{3}\text{e}^{3\text{x}}+\text{c}$
View full question & answer→Question 132 Marks
Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\text{s}^2\frac{\text{d}^2\text{t}}{\text{ds}^2}+\text{st}\frac{\text{dt}}{\text{ds}}=\text{s}$
Answer$\text{s}^2\frac{\text{d}^2\text{t}}{\text{ds}^2}+\text{st}\frac{\text{dt}}{\text{ds}}=\text{s}$ $\Rightarrow\text{s}\frac{\text{d}^2\text{t}}{\text{ds}}^2+\text{t}\frac{\text{dt}}{\text{ds}}=1$In this differential equation, the order of the highest order derivative is 2 and its power is 1. So, it is a differential equation of order 2 and degree 1.
It is a non-linear differential equation, as it contains the product of the dependent variable (t) and its differential co-efficient $\Big(\frac{\text{dt}}{\text{ds}}\Big).$
View full question & answer→Question 142 Marks
Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\frac{\text{d}^2\text{y}}{\text{dx}^2}+4\text{y}=0$
Answer$\frac{\text{d}^2\text{y}}{\text{dx}^2}+4\text{y}=0$
It is a linear differential equation.
The highest order differential coefficient is $\frac{\text{d}^2\text{y}}{\text{dx}^2}$ and its power is 1.
So, It is a linear differential equation with order 2 and degree 1.
View full question & answer→Question 152 Marks
Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\sqrt{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2}=\Big(\text{c}\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)^{\frac{1}{3}}$
Answer$\sqrt{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2}=\Big(\text{c}\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)^{\frac{1}{3}}$ Squaring both sides, we get $\Rightarrow1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2=\Big(\text{c}\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)^{\frac{2}{3}}$ Taking cubes of both sides, we get $\Rightarrow\Big(\text{c}\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)=\Big[1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2\Big]^3$ $\Rightarrow\Big(\text{c}\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)=1+3\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+3\Big(\frac{\text{dy}}{\text{dx}}\Big)^4+\Big(\frac{\text{dy}}{\text{dx}}\Big)^6$In this differential equation, the order of the highest order derivative is 2 and its power is 2. So, it is a differential equation of order 2 and degree 2.
It is a non-linear differential equation, as its degree is more than 1.
View full question & answer→Question 162 Marks
Solve the following differential equations:
$(\text{y}^2+1)\text{dx}-(\text{x}^2+1)\text{dy}=0$
Answer$(\text{y}^2+1)\text{dx}-(\text{x}^2+1)\text{dy}=0$
$(\text{y}^2+1)\text{dx}=(\text{x}^2+1)\text{dy}$
$\int\frac{\text{dy}}{\text{y}^2+1}=\int\frac{\text{dx}}{\text{x}^2+1}$
$\tan^{-1}\text{y}=\tan^{-1}\text{x + C}$
View full question & answer→Question 172 Marks
Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\frac{\text{d}^3\text{x}}{\text{dt}^3}+\frac{\text{d}^3\text{x}}{\text{dt}^2}+\Big(\frac{\text{ dx}}{\text{dt}}\Big)^2=\text{e}^\text{t}$
AnswerIn this differential equation, the order of the highest order derivative is 3 and its power is 1. So, it is a differential equation of order 3 and degree 1.
It is a non-linear differential equation because the differential coefficient $\frac{\text{dx}}{\text{dt}}$ has exponent 2, which is greater than 1.
View full question & answer→Question 182 Marks
Define a differential equation.
AnswerAn equation containing an independent variable, a dependent variable and differential cofficients of the dependent variable with reapect to the independent variable is called a differential equation.
for example: $\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}+\text{y}}$
View full question & answer→Question 192 Marks
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}+\text{y}}+\text{e}^\text{y}\text{x}^3$
Answer$\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}+\text{y}}+\text{e}^\text{y}\text{x}^3$
$\frac{\text{dy}}{\text{dx}}=\text{e}^\text{y}(\text{e}^\text{x}+\text{x}^3)$
$\int\text{e}^{-\text{y}}\text{dy}=\int(\text{e}^\text{x}+\text{x}^3)\text{dx}$
$-\text{e}^{-\text{y}}-\text{e}^\text{x}+\frac{\text{x}^4}{4}+\text{C}_1$
$\text{e}^\text{x}+\frac{\text{x}^4}{4}+\text{e}^{-\text{y}}=\text{C}$
View full question & answer→Question 202 Marks
Write the order of the differential equation $\text{y}=\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\sqrt{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^{2}}.$
AnswerWe have,
$\text{y}=\text{x}\frac{\text{dy}}{\text{dx}}+\text{a}\sqrt{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^{2}}$
$\text{y}-\text{x}\frac{\text{dy}}{\text{dx}}=\text{a}\sqrt{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^{2}}$
Squaring both sides, we get
$\Big(\text{y}-\text{x}\frac{\text{dy}}{\text{dx}}\Big)^{2}=\left\{\text{a}\sqrt{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^{2}}\right\}^{2}$
$\text{y}^{2}-\text{x}^{2}\Big(\frac{\text{dy}}{\text{dx}}\Big)^{2}-2\text{xy}\frac{\text{dy}}{\text{dx}}=\text{a}^{2}\left\{{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^{2}}\right\}^{2}$
$\text{y}^{2}+\Big(\frac{\text{dy}}{\text{dx}}\Big)^{2}(\text{x}^{2}-\text{a}^{2})-2\text{xy}\frac{\text{dy}}{\text{dx}}=\text{a}^{2}$
From the above equation, we see that the highest order is 1.
So, its order is 1 and the power of the order is 2.
Thus, it is differential equation of order 1 and degree 2.
View full question & answer→Question 212 Marks
Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\frac{\text{d}^2\text{y}}{\text{dx}^2}+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+\text{xy}=0$
Answer$\frac{\text{d}^2\text{y}}{\text{dx}^2}+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+\text{xy}=0$ In this differential equation, the order of the highest order derivative is 2 and its power is 1. So, it is a differential equation of order 2 and degree 1.It is a non-linear differential equation, as the differential coefficient $\frac{\text{dy}}{\text{dx}}$ has exponent 2, which is greater than 1.
View full question & answer→Question 222 Marks
For the following differntial equations verify that the accompanying function is a solution:
| Differential equation |
Function |
| $\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}$ |
$\text{y}=\text{ax}$ |
AnswerWe have $\text{y}=\text{ax}\ ...(1)$ Given differential equation Differentiating both sides of (1) with respect to x, we get $\frac{\text{dy}}{\text{dx}}=\text{a}$ $\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}$ $\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}={\text{y}}$Hence, the given function is the solution to the given differential equation.
View full question & answer→Question 232 Marks
Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
${3}\sqrt{\frac{\text{d}^2\text{y}}{\text{dx}^2}}=\sqrt{\frac{\text{dy}}{\text{dx}}}$
Answer${3}\sqrt{\frac{\text{d}^2\text{y}}{\text{dx}^2}}=\sqrt{\frac{\text{dy}}{\text{dx}}}$ $\Rightarrow\Big(\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)^{\frac{1}{3}}=\Big(\frac{\text{dy}}{\text{dx}}\Big)^\frac{1}{2}$ Taking cubes of both the sides, we get $\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=\Big(\frac{\text{dy}}{\text{dx}}\Big)^{\frac{3}{2}}$ Squaring both the sides, we get $\Rightarrow\Big(\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)^2=\Big(\frac{\text{dy}}{\text{dx}}\Big)^3$ $\Rightarrow\Big(\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)^2-\Big(\frac{\text{dy}}{\text{dx}}\Big)^3=0$In this differential equation, the order of the highest order derivative is 2 and its power is 2. So, it is a differential equation of order 2 and degree 2.
Thus, it is a non-linear differential equation, as its degree is 2, which is greater than 1.
View full question & answer→Question 242 Marks
Show that $\text{y}=\text{e}^{-\text{x}}+\text{ax}+\text{b}$ is solution of the differential equation $\text{e}^\text{x}\frac{\text{d}^2\text{y}}{\text{dx}^2}=1$
AnswerWe have $\text{y}=\text{e}^{-\text{x}}+\text{ax}+\text{b}\ ...(1)$ Differentiating both sides of (1) with respect to x, we get $\frac{\text{dy}}{\text{dx}}=-\text{e}^{-\text{x}}+\text{a}\ ...(2)$ Differentiating both sides of (1) with respect to x, we get $\frac{\text{d}^2\text{y}}{\text{dx}^2}=\text{e}^{-\text{x}}$ $\Rightarrow\text{e}^\text{x}\frac{\text{d}^2\text{y}}{\text{dx}^2}=1$Hence, the given function is the solution to the given differential equation.
View full question & answer→Question 252 Marks
Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\text{y}\frac{\text{d}^2\text{x}}{\text{dy}^2}=\text{y}^2+1$
Answer$\text{y}\frac{\text{d}^2\text{x}}{\text{dy}^2}=\text{y}^2+1$ $\frac{\text{d}^2\text{x}}{\text{dy}^2}-\text{y}-\frac{1}{\text{y}}=0$The differential coefficient is $\frac{\text{d}^2\text{x}}{\text{dy}^2}$ and its power is 1.
So, it is a linear differential equation with order 2 and degree 1.
View full question & answer→Question 262 Marks
Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$2\frac{\text{d}^2\text{y}}{\text{dx}^2}+3\sqrt{1-\Big(\frac{\text{dy}}{\text{dx}}\Big)^2-\text{y}}=0$
Answer$2\frac{\text{d}^2\text{y}}{\text{dx}^2}+3\sqrt{1-\Big(\frac{\text{dy}}{\text{dx}}\Big)^2-\text{y}}=0$
$2\frac{\text{d}^2\text{y}}{\text{dx}^2}+3\sqrt{1-\Big(\frac{\text{dy}}{\text{dx}}\Big)^2-\text{y}}$
Squaring both the sides,
$4\Big(\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)^2=9\Big(1-\Big(\frac{\text{dy}}{\text{dx}}\Big)^2-\text{y}\Big)$
$4\Big(\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)^2+9\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+9\text{y}-9=0$
The highest order differential coefficient is $\frac{\text{d}^2\text{y}}{\text{dx}^2}$ and its power is 2.
So, it is a non linear differential equation with order 2 and degee 2.
View full question & answer→Question 272 Marks
Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$5\frac{\text{d}^2\text{y}}{\text{dx}^2}=\Big\{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2\Big\}^{\frac{3}{2}}$
Answer$5\frac{\text{d}^2\text{y}}{\text{dx}^2}=\Big\{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2\Big\}^{\frac{3}{2}}$
$\Big\{5\Big(\frac{\text{d}^2\text{y}}{\text{dx}}\Big)^2\Big\}=\Big\{1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^2\Big\}^3$
$25\Big(\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)^2=1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^6+3\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+3\Big(\frac{\text{dy}}{\text{dx}}\Big)^4$
$25\Big(\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)^2-\Big(\frac{\text{dy}}{\text{dx}}\Big)^6-3\Big(\frac{\text{dy}}{\text{dx}}\Big)^4-3\Big(\frac{\text{dy}}{\text{dx}}\Big)^2-1=0$
The highest order differential coefficient is $\frac{\text{d}^2\text{y}}{\text{dx}^2}$ and its power is 2.
So, it is a non linear differential equation with order 2 and degee 2.
View full question & answer→Question 282 Marks
Represent the following families of curves by forming the corresponding differential equation:
$\text{x}^2-\text{y}^2=\text{a}^2$
AnswerThe equation of the family of curves is
$\text{x}^2-\text{y}^2=\text{a}^2\ ...(1)$
where a is a parameter.
As this equation has only one arbitrary constant, we shall get a differential equation of first order.
Differentiating (1) with respect to x, we get
$2\text{x}-2\text{y}\frac{\text{dy}}{\text{dx}}=0$
$\Rightarrow\text{x}-\text{y}\frac{\text{dy}}{\text{dx}}=0$
It is the required differential equation.
View full question & answer→Question 292 Marks
Represent the following families of curves by forming the corresponding differential equation:
$\text{x}^2+\text{y}^2=\text{a}^2$
AnswerThe equation of the family of curves is
$\text{x}^2+\text{y}^2=\text{a}^2$
where a is a parameter.
As this equation has only one arbitrary constant, we shall get a differential equation of first order.
Differentiating (1) with respect to x, we get
$2\text{x}+2\text{y}\frac{\text{dy}}{\text{dx}}=0$
$\Rightarrow\text{x}+\text{y}\frac{\text{dy}}{\text{dx}}=0$
It is the required differential equation.
View full question & answer→