True False[1 Marks ]

Question 11 Mark

State True or False for the following:
The differential equation representing the family of circles $x^2+(y-a)^2=a^2$ will be of order two.

Answer

False.
Solution:
We know that, order of a differential equation = Number of arbitrary constants
Here, number of arbitrary constants = 1
So the required order of differential equation is one.

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Question 21 Mark

State True or False for the following:
The solution of the differential equation $\frac{\text{dy}}{\text{dx}}=\frac{\text{x}+2\text{y}}{\text{x}}$ is $\text{x}+\text{y}=\text{k}\text{x}^2.$

Answer

True.Solution:
We have
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x}+2\text{y}}{\text{x}}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=1+\frac{2}{\text{x}}.\text{y}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}-\frac{2}{\text{x}}=\text{y}$
This is a linear differential equation.
$\therefore\text{I.F.}=\text{e}^{\frac{-2}{\text{x}}\text{dx}}$
$=\text{e}^{-2\log\text{x}}=\text{x}^{-2}$
Thus, the differential solution is given as,
$\text{y}.\text{x}^{-2}=\int\text{x}^{-2}.1\text{dx}+\text{k}$
$\Rightarrow\frac{\text{y}}{\text{x}^2}=\frac{\text{x}^{-2+1}}{-2+1}+\text{k}$
$\Rightarrow\frac{\text{y}}{\text{x}^2}=\frac{-1}{\text{x}}+\text{k}$
$\Rightarrow\text{y}=-\text{x}+\text{k}\text{x}^2$
$\Rightarrow\text{y}+\text{x}=\text{k}\text{x}^2$

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Question 31 Mark

State True or False for the following:
Solution of $\frac{\text{xdy}}{\text{dx}}=\text{y}+\text{x}\tan\Big(\frac{\text{y}}{\text{x}}\Big)$ is $\sin\Big(\frac{\text{y}}{\text{x}}\Big)=\text{cx}.$

Answer

True.Solution:
We have
$\frac{\text{xdy}}{\text{dx}}=\text{y}+\text{x}\tan\Big(\frac{\text{y}}{\text{x}}\Big)$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}+\text{x}\tan\Big(\frac{\text{y}}{\text{x}}\Big)\ .....(\text{i})$
Put $\frac{\text{y}}{\text{x}}=\text{v}$ or $\text{y}=\text{vx}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{v}+\frac{\text{dv}}{\text{dx}}$
On substituting these values in Eq. (i), we get
$\frac{\text{xdy}}{\text{dx}}+\text{v}=\text{v}+\tan\text{v}$
$\Rightarrow\frac{\text{dx}}{\text{x}}=\frac{\text{dv}}{\tan\text{v}}$
On integrating both sides, we get
$\int\frac{1}{\text{x}}\text{dx}=\int\cot\text{v}\text{dv}$
$\Rightarrow\log\text{x}+\log\text{C}=\log\sin\text{v}$
$\Rightarrow\log\text{Cx}=\log\sin\text{v}$
$\Rightarrow\sin\text{v}=\text{Cx}$
$\Rightarrow\sin\frac{\text{y}}{\text{x}}=\text{C}\text{x}^4$

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Question 41 Mark

State True or False for the following:
The differential equation of all non horizontal lines in a plane is $\frac{\text{d}^2\text{x}}{\text{d}\text{y}^2}=0.$

Answer

True.Solution:
Let any non-horizontal line in a plane is given by
$\text{y}=\text{mx}+\text{c}$
On differentiating w.r.t.x, we get
$\frac{\text{dy}}{\text{dx}}=\text{m}$
Again, differentiating w.r.t.x, we get
$\frac{\text{d}^2\text{x}}{\text{d}\text{y}^2}=0$

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Question 51 Mark

State True or False for the following:
Number of arbitrary constants in the particular solution of a differential equation of order two is two.

Answer

False.
Solution:
There is no arbitrary constant in the particular solution of a differential equation.

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Question 61 Mark

State True or False for the following:
Differential equation representing the family of curves $\text{y}=\text{e}^{\text{x}}(\text{A}\cos\text{x}+\text{B}\sin\text{x})$ is $\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}-2\frac{\text{dy}}{\text{dx}}+2\text{y}=0.$

Answer

True.Solution:
We have $\text{y}=\text{e}^{\text{x}}(\text{A}\cos\text{x}+\text{B}\sin\text{x})$
On differentiating w.r.t.x we get
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}}(-\text{A}\sin\text{x}+\text{B}\sin\text{x})+\text{e}^{\text{x}}(\text{A}\cos\text{x}+\text{B}\sin\text{x})$
Again differentiating w.r.t.x we get
$\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}-\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}}(-\text{A}\cos\text{x}-\text{B}\sin\text{x})+\text{e}^{\text{x}}(-\text{A}\sin\text{x}+\text{B}\cos\text{x})$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}-\frac{\text{dy}}{\text{dx}}+\text{y}=\frac{\text{dy}}{\text{dx}}-1$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{d}\text{x}^2}-2\frac{\text{dy}}{\text{dx}}+2\text{y}=0$

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Question 71 Mark

State True or False for the following:
Solution of the differential equation of the type $\frac{\text{dy}}{\text{dx}}+\text{P}_1\text{x}=\text{Q}_1$ is given by $\text{x.I.F.}=\int(\text{I.F.})\times\text{Q}_1\text{dy}.$

Answer

True.

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Question 81 Mark

State True or False for the following:
Correct substitution for the solution of the differential equation of the type $\frac{\text{dy}}{\text{dx}}=\text{g}(\text{x, y})$ where g(x, y) is a homogeneous function of the degree zero is x = vy.

Answer

True.

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Question 91 Mark

State True or False for the following:
Integrating factor of the differential of the form $\frac{\text{dy}}{\text{dx}}+\text{P}_1\text{x}=\text{Q}_1$ is given by $\text{e}^{\text{P}_1\text{dy}}.$

Answer

True.Solution:
Given differential equation,
$\frac{\text{dy}}{\text{dx}}+\text{P}_1\text{x}=\text{Q}_1$
$\text{I.F.}=\int\text{e}^{\text{P}_1\text{dy}}$

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Question 101 Mark

State True or False for the following:
Correct substitution for the solution of the differential equation of the type $\frac{\text{dy}}{\text{dx}}=({\text{x}},\text{y}),$ where f(x, y) is a homogeneous function of zero degree is y = vx.

Answer

True.

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Question 111 Mark

State True or False for the following:
The solution of $\frac{\text{dy}}{\text{dx}}=\Big(\frac{\text{y}}{\text{x}}\Big)^{\frac{1}{3}}$ is $\text{y}^{\frac{2}{3}}-\text{x}^{\frac{2}{3}}=\text{C}.$

Answer

True.Solution:
We have $\frac{\text{dy}}{\text{dx}}=\frac{\text{y}^{\frac{1}{3}}}{\text{x}^{\frac{1}{3}}}$
$\Rightarrow\text{y}^{-\frac{1}{3}}\text{dy}=\text{x}^{-\frac{1}{3}}\text{dx}$
$\Rightarrow\int\text{y}^{-\frac{1}{3}}\text{dy}=\int\text{x}^{-\frac{1}{3}}\text{dx}$
$\Rightarrow\frac{3}{2}\text{y}^{\frac{2}{3}}=\frac{3}{2}\text{x}^{\frac{2}{3}}+\text{C}'$
$\Rightarrow\text{y}^{\frac{2}{3}}-\text{x}^{\frac{2}{3}}=\text{C}$

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