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Question 511 Mark

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\frac{\text{d}^2\text{y}}{\text{dx}^2}+3\Big(\frac{\text{dy}}{\text{dx}}\Big)^2=\text{x}^2\log\Big(\frac{\text{d}^2\text{y}}{\text{dx}^2}\Big)$

Answer

$\frac{\text{dy}}{\text{dx}}+\text{e}^\text{x}=0$
The highest order differential coefficient is $\frac{\text{dy}}{\text{dx}}$ and its power is 1.
So, it is a non linear differential equation of order 1 and degree 1.

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Question 521 Mark

Determine order and degree $($if defined$)$ of differential equations given in Exercise. $y\ ' + y = e^x$

Answer

The given differential equation is $y\ ' + y = e^x$
The highest order derivative present in the given differential equation is $y\ '$ and index of its highest power is $1.$
$\therefore$ the given differential equation is of order $1$ and degree $1.$

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Question 531 Mark

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\frac{\text{d}^3\text{y}}{\text{dx}^3}+\frac{\text{d}^2\text{y}}{\text{dx}^2}+\frac{\text{dy}}{\text{dx}}+\text{y}\sin\text{y}=0$

Answer

$\frac{\text{d}^3\text{y}}{\text{dx}^3}+\frac{\text{d}^2\text{y}}{\text{dx}^2}+\frac{\text{dy}}{\text{dx}}+\text{y}\sin\text{y}=0$The highest order differential coefficient is $\frac{\text{d}^2\text{y}}{\text{dx}^2}$ and its power is 1.
So, it is non linear differential equation with order 3 and degree 1.

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Question 541 Mark

Determine the order and degree of the following differential equations. state also whether they are linear or non linear. $\frac{\text{dy}}{\text{dx}}+\text{e}^\text{y}=0$

Answer

$\frac{\text{dy}}{\text{dx}}+\text{e}^\text{y}=0$ In this differential equation, the order of the highest order derivative is $1$ and its power is $1.$ So, it is a differential equation of order $1$ and degree $1.$
It is a non$-$linear differential equation, as the exponent of the dependent variable is not equal to $1($as per expansion series series of $e^y).$

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Question 551 Mark

Write the degree of the following differrntial equation $ \Big(\frac{\text{dy}}{\text{dx}}\Big)^{4}+3\text{x}\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=0.$

Answer

$ \Big(\frac{\text{dy}}{\text{dx}}\Big)^{4}+3\text{x}\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=0$
Here, we see that the highest order derivative is $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}$ and its power is 1.
Therefore, the given differential equation is of first degree.

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Question 561 Mark

Write the differential equation representing the famliy of straight line y = Cx + 5, where C is an arbitrary constant.

Answer

We have,
$\text{y}=\text{Cx}+5\ ...(\text{i})$
$\Rightarrow \frac{\text{dy}}{\text{dx}}=\text{C}$
Substituting the value of C in (i),
$\text{y}=\frac{\text{dy}}{\text{dx}}\times\text{x}+5$
$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}+5=0$
Hence, $\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}+5=0$ is the differential equation tha famliy of lines $\text{y}=\text{Cx}+5$ When C is an arbita.

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Question 571 Mark

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\Big(\frac{\text{dy}}{\text{dx}}\Big)^3-4\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+7\text{y}=\sin\text{x}$

Answer

$\Big(\frac{\text{dy}}{\text{dx}}\Big)^3-4\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+7\text{y}=\sin\text{x}$The order of a differential equation is the order of the highest order derivative appearing in the equation. The degree of a differential equation is the degree of the highest order derivative. Consider the given differential equation
$\Big(\frac{\text{dy}}{\text{dx}}\Big)^3-4\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+7\text{y}=\sin\text{x}$
In the above equation, the order of the highest order derivative is 1. So the differential equation is of order 1. In the above differential equation, the power of the highest order derivative is 3. Hence, it is a differential equation of degree 3. Since the degree of the above differential equation is 3, more than one, it is a non-linear differential equation.

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Question 581 Mark

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\text{x}^2\Big(\frac{\text{d}^2\text{y}}{\text{dx}}\Big)^3+\text{y}\Big(\frac{\text{dy}}{\text{dx}}\Big)^4+\text{y}^4=0$

Answer

$\text{x}^2\Big(\frac{\text{d}^2\text{y}}{\text{dx}}\Big)^3+\text{y}\Big(\frac{\text{dy}}{\text{dx}}\Big)^4+\text{y}^4=0$In this differential equation, the order of the highest order derivative is 2 and its power is 3. So, it is a differential equation of order 2 and degree 3.
It is a non linear differential equation, as its degree is more than 1.

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Question 591 Mark

Write the order of the differential equation of the famliy of circles of radius r.

Answer

Given, the equation of famliy of circle,
$(\text{x}-\text{a}^{2})+(\text{y}-\text{b})^{2}=\text{r}^{2}\ ...(\text{i})$
Since, given equation have two arbiteary constant, so we differential the above equation two times wrt.x.
Differential equation (i) wrt.x. we get
$2(\text{x}-\text{a})+2(\text{y}-\text{b})\frac{\text{}dy}{\text{dx}}=\text{r}^{2}=0$
$\Rightarrow (\text{x}-\text{a})+(\text{y}-\text{b})\frac{\text{dy}}{\text{dx}}=0\ ...(\text{ii})$
Differential equation (ii) wrt.x. we get,
$1+(\text{y}-\text{b})\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}+\big(\frac{\text{dy}}{\text{dx}}\big)=0$
$\Rightarrow(\text{y}-\text{b})=\frac{1+\big(\frac{\text{dy}}{\text{dx}}\Big)^{2}}{\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}}\ ...(\text{iii})$
Substitite value of (y - b) in equation (ii), we have,
$(\text{x}-\text{a})+\Bigg[\frac{1+\big(\frac{\text{dy}}{\text{dx}}\Big)^{2}}{\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}}\Bigg]\frac{\text{dy}}{\text{dx}}=0$
$(\text{x}+\text{a})+\frac{\Big[ 1+\big(\frac{\text{dy}}{\text{dx}}\Big)^{2}\Big]\frac{\text{dy}}{\text{dx}}}{\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}}\ ...(\text{iv})$
Substitite value of (x - a) (y - b) in equation (ii), we have,
$\Bigg[\frac{1+\big(\frac{\text{dy}}{\text{dx}}\Big)^{2}}{\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}}\Bigg]\frac{\text{dy}}{\text{dx}}+\Bigg[\frac{1+\big(\frac{\text{dy}}{\text{dx}}\Big)^{2}}{\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}}\Bigg]\frac{\text{dy}}{\text{dx}} =\text{r}^{2}$
$\Rightarrow \Big[1+\big(\frac{\text{dy}}{\text{dx}}\big)^{2}\Big] ^{3}=\text{r}^{2}\Big(\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}\Big)^{2}$
The order of the differential equation of the famliy of circle of redius r is 2.

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Question 601 Mark

Determine order and degree (if defined) of differential equations given in Exercise.
y''' + 2y" + y' = 0

Answer

The highest order derivative present in the given differential equation is y" and index of its highest power is 1.
$\therefore$ the given differential equation is of order 3 and degree 1.

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MCQ 611 Mark

Family $y = Ax + A^4$ of curves is represented by the differential equation of degree$:$

A
$3$
B
$2$
C
$4$
✓
$1$

Answer

Correct option: D.

$1$

$Y = Ax + A^4$
This equation is a linear Differential equation $=\frac{\text{dy}}{\text{dx}}=\text{A}$
Here the highest order Derivative is $y$
The Degree of this Derivative is $1$

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Question 621 Mark

Write the order of the differential equation of the family of circles touching X-axis at the origin.

Answer

The equation of the famliy of circles touching x-axis at the origin is
$(\text{x}-0)^{2}+(\text{y}-\text{a})^{2}=\text{a}^{2}$
$\text{x}^{2}+\text{y}^{2}-2\text{ay}=0\ ...(\text{i})$
Here, a is the parameter.
Since, this equation contain only one conatant, we differentiate it only once.
$2\text{x}+2\text{y}\frac{\text{dy}}{\text{dx}}-2\text{a}\frac{\text{dy}}{\text{dx}}=0$
$\text{a}=\frac{\text{x}+\text{y}(\frac{\text{dy}}{\text{dx}})}{\frac{\text{dy}}{\text{dx}}}\ ...(\text{ii})$
Putting the value of a from (i) in (i), we get
$\text{x}^{2}+\text{y}^{2}=2\text{y}\left\{\frac{\text{x}+\text{y}(\frac{\text{dy}}{\text{dx}})}{\frac{\text{dy}}{\text{dx}}}\right\}$
$(\text{x}^{2}+\text{y}^{2})\frac{\text{dy}}{\text{dx}}=2\text{xy}$
So, this is the differential equation.
Here, order of the diffrential equation is 1.

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Question 631 Mark

How many arbitray constants are there in the genral solution of the differential equation of orader 3.

Answer

The arbitrary constants in the general solution of the differential equation is equal to the order of the differntial equation.
Hence, the number of arbitrary constant in the general solution of the order 3 are 3.

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MATHS 1 Marks Question