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Differential Equations · Gujarat Board (GSEB) STD 12 Science (GSEB - English Medium). 28 questions.

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Question 11 Mark
The general solution of the differential equation $\frac{y d x-x d y}{y}=0$ is 
Answer
It is given that $\frac{\mathrm{ydx}-\mathrm{xdy}}{\mathrm{y}}=0$ 
$\Rightarrow \frac{\mathrm{ydx}-\mathrm{xdy}}{\mathrm{xy}}=0$ 
$\Rightarrow \frac{1}{x} d x-\frac{1}{y} d y=0$ 
Integrating both sides, we get,
log|x| - log|y| = log k
$\Rightarrow \log \left|\frac{x}{y}\right|=\log k$ 
$\Rightarrow \frac{x}{y}=k$ 
$\Rightarrow \mathrm{y}=\frac{1}{\mathrm{k}} \mathrm{x}$ 
$\Rightarrow$ y = Cx where C = $\frac{1}{k}$ 
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Question 21 Mark
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
Answer
It is given that exdy + (yex + 2x) dx = 0
$\Rightarrow \mathrm{e}^{\mathrm{x}} \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{ye}^{\mathrm{x}}+2 \mathrm{x}=0$ 
$\Rightarrow \frac{d y}{d x}+y=-2 x e^{-x}$ 
This is equation in the form of $\frac{d y}{d x}+p y=Q$ (where, p = 1 and Q = -2xe-x)
Now, I.F = $\mathrm{e}^{\int \mathrm{pdx}}=\mathrm{e}^{\int \mathrm{dx}}=\mathrm{e}^{\mathrm{x}}$ 
Thus, the solution of the given differential equation is given by the relation:
$y(I . F .)=\int(Q \times I . F .) d x+C$ 
$\Rightarrow \mathrm{ye}^{\mathrm{x}}=\int\left(-2 \mathrm{xe}^{-\mathrm{x}} \cdot \mathrm{e}^{\mathrm{x}}\right) \mathrm{dx}+\mathrm{C}$ 
$\Rightarrow \mathrm{ye}^{\mathrm{x}}=-\int 2 \mathrm{xdx}+\mathrm{C}$ 
$\Rightarrow$ yex = -x2 + C
$\Rightarrow$ yex + x2 = C
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Question 31 Mark
The general solution of a differential equation of the type $\frac{d x}{d y}+\mathrm{P}_{1} x=\mathrm{Q}_{1}$ is
Answer
The integrating factor of the given differential equation
$\frac{\mathrm{d} \mathrm{x}}{\mathrm{dy}}+\mathrm{P}_{1} \mathrm{x}=\mathrm{Q}_{1} \text { is } \mathrm{e}^{\int \mathrm{P}_{1} \mathrm{d} \mathrm{y}}$ 
Thus, the general solution of the differential equation is given by,
$\mathrm{x}(\mathrm{I.F.})=\int(\mathrm{Q_1} \times \mathrm{I.F.}) \mathrm{dy}+\mathrm{C}$ 
$\Rightarrow \mathrm{x} . \mathrm{e}^{\int \mathrm{P}_{1} \mathrm{d} \mathrm{y}}=\int\left(\mathrm{Q}_{1} \mathrm{e}^{\int \mathrm{P}_{1} \mathrm{d} \mathrm{y}}\right) \mathrm{d} \mathrm{y}+\mathrm{C}$
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Question 41 Mark
Find the order and degree (if defined) of the differential equation $\frac{d^{4} y}{d x^{4}}-\sin \left(\frac{d^{3} y}{d x^{3}}\right)=0$
Answer
It is given that the equation is $\frac{d^{4} y}{d x^{4}}-\sin \left(\frac{d^{3} y}{d x^{3}}\right)=0$
We can see that the highest order derivative present in the differential is $\frac{d^{4} y}{d x^{4}}$.
Thus, its order is four.
The given differential equation is not a polynomial equation.
Therefore, its degree is not defined.
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Question 51 Mark
Find the order and degree (if defined) of the differential equation $\left(\frac{d y}{d x}\right)^{3}-4\left(\frac{d y}{d x}\right)^{2}+7 y=\sin x$
Answer
It is given that equation is $\left(\frac{d y}{d x}\right)^{3}-4\left(\frac{d y}{d x}\right)^{2}+7 y=\sin x$ 
$\left(\frac{d y}{d x}\right)^{3}-4\left(\frac{d y}{d x}\right)^{2}+7 y-\sin x=0$ 
We can see that the highest order derivative present in the differential is $\frac{dy}{dx}$.
Thus, its order is one. It is polynomial equation in $\frac{dy}{dx}$. The highest power raised to $\frac{dy}{dx}$ is 3.
Therefore, its degree is three.
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Question 61 Mark
Find the order and degree (if defined) of the differential equation $\frac{d^{2} y}{d x^{2}}+5 x\left(\frac{d y}{d x}\right)^{2}-6 y=\log x$
Answer
It is given that equation is $\frac{d^{2} y}{d x^{2}}+5 x\left(\frac{d y}{d x}\right)^{2}-6 y=\log x$ 
$\Rightarrow\frac{d^{2} y}{d x^{2}}+5 x\left(\frac{d y}{d x}\right)^{2}-6 y-\log x=0$ 
We can see that the highest order derivative present in the differential is $\frac{d^{2} y}{d x^{2}}$.
Thus, its order is two.
The highest power raised to $\frac{d^{2} y}{d x^{2}}$ is 1.
Therefore, its degree is one.
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Question 71 Mark
The Integrating Factor of the differential equation $\left(1-y^{2}\right) \frac{d x}{d y}+y x=a y~(-1<y<1)$ is
Answer
It is given that $\left(1-y^{2}\right) \frac{d x}{d y}+y x=a y$ 
$\Rightarrow \frac{d x}{d y}+\frac{y x}{1-y^{2}}=\frac{a y}{1-y^{2}}$ 
This is equation in the form of $\frac{d x}{d y}+p x=Q$ (where $p=\frac{y}{1-y^{2}}$ and Q = $\frac{ay}{1-y^{2}}$)
Now, I.F. = $\mathrm{e}^{\int \mathrm{p} \mathrm{d} \mathrm{y}}=\mathrm{e}^{\int \frac{\mathrm{y}}{1-\mathrm{y}^{2}} \mathrm{d} \mathrm{y}}=\mathrm{e}^{-\frac{1}{2} \log \left(1-\mathrm{y}^{2}\right)}=\mathrm{e}^{\log \left[\frac{1}{\sqrt{\left(1-\mathrm{y}^{2}\right)}}\right]}$ 
= $\frac{1}{\sqrt{\left(1-y^{2}\right)}}$
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Question 81 Mark
The Integrating Factor of the differential equation $x \frac{d y}{d x}-y=2 x^{2}$ is 
Answer
It is given that $x \frac{d y}{d x}-y=2 x^{2}$ 
$\Rightarrow \frac{d y}{d x}-\frac{y}{x}=2 x$ 
This is equation in the form of $\frac{d y}{d x}+p y=Q$ (where, $p=-\frac{1}{x}$ and Q = 2x2)
Now, I.F. = $e^{\int p d x}=e^{\int \frac{-1}{x} d x}=e^{\log \left(x^{-1}\right)}=x^{-1}=\frac{1}{x}$
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Question 91 Mark
Which of the following is a homogeneous differential equation?
Answer
it  is a homogeneous differential equation ,because the degree of each individual term is same i.e. 2.
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Question 101 Mark
A homogeneous equation of the form $\frac{{dy}}{{dx}} = h\left( {\frac{x}{y}} \right)$ can be solved by making the substitution
Answer
A homogeneous equation of the form $\frac{{dy}}{{dx}} = h\left( {\frac{x}{y}} \right)$ can be solved by making the substitution x= vy.so that it becomes variable separable form and integration is then possible
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Question 111 Mark
The general solution of the differential equation $\frac{d y}{d x}=e^{x+y}$ is
Answer
We have, $ \frac{d y}{d x}=e^{x+y}$ 
$\Rightarrow \frac{d y}{d x}=e^{x} \times e^{y}$ 
separating variables
$\Rightarrow \mathrm{e}^{-\mathrm{y}} \mathrm{dy}=\mathrm{e}^{\mathrm{x}} \mathrm{d} \mathrm{x}$ 
Integrating both sides
$\Rightarrow \int \mathrm{e}^{-\mathrm{y}} \mathrm{d} \mathrm{y}=\int \mathrm{e}^{\mathrm{x}} \mathrm{d} \mathrm{x}$ 
$\Rightarrow-e^{-y}=e^{x}+c$ 
$\Rightarrow e^{x}+e^{-y}=-c$ 
Or,
$e^{x}+e^{-y}=c$ (c is a constant)
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Question 121 Mark
The number of arbitrary constants in the particular solution of a differential equation of third order are:
Answer
0, because the particular solution is free from arbitrary constants.
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Question 131 Mark
The number of arbitrary constants in the general solution of a differential equation of fourth order are:
Answer
4, because the no. of arbitrary constants is equal to order of the differential equation.
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Question 141 Mark
Determine order and degree (if defined) of differential equation: y" + (y')2 + 2y = 0 
Answer
It is given that equation is y’’ + (y’)2 + 2y = 0
We can see that the highest order derivative present in the differential is y”.
Thus, its order is two.
Highest power raised to y” is 1
Therefore, its degree is one.
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Question 151 Mark
Determine order and degree (if defined) of differential equation: y' + y = ex
Answer
It is given that equation is y’ + y = ex
$\Rightarrow$ y’ + y – ex = 0
We can see that the highest order derivative present in the differential is y’.
Thus, its order is one. It is polynomial equation in y’
So, the highest power raised to y’ is 1
Therefore, its degree is one.
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Question 161 Mark
Determine order and degree (if defined) of differential equation y'" + 2y" + y' = 0
Answer
It is given that equation is y″′ +2y” + y’ = 0
We can see that the highest order derivative present in the differential is y”’.
Thus, its order is three. It is polynomial equation in y”’, y” and y’
So, the highest power raised to y”’ is 1
Therefore, its degree is one.
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Question 171 Mark
Determine order and degree (if defined) of differential equation $\left(y^{\prime \prime \prime}\right)^{2}+\left(y^{\prime \prime}\right)^{3}+\left(y^{\prime}\right)^{4}+y^{5}=0$
Answer
It is given that equation is (y”’)2 + (y”)3 +(y’)4 + y5 = 0
We can see that the highest order derivative present in the differential is y”’.
Thus, its order is three.
It is polynomial equation in y”’, y” and y’
So, the highest power raised to y”’ is 2
Therefore, its degree is two.
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Question 181 Mark
Determine order and degree (if defined) of differential equation $\frac{d^{2} y}{d x^{2}}$ = cos 3x + sin 3x
Answer
The given differential equation is  $\frac{d^{2} y}{d x^{2}}$ = cos 3x + sin 3x
$\Rightarrow \frac{d^{2} y}{d x^{2}}$ - cos 3x - sin 3x = 0
We can see that the highest order derivative present in the given differential equation is $\frac{d^{2} y}{d x^{2}}$
Thus, its order is two.
It is polynomial equation in $\frac{d^{2} s}{d t^{2}}$ and the power is 1
Therefore, its degree is one.
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Question 191 Mark
Determine order and degree (if defined) of differential equation $\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\cos \left(\frac{d y}{d x}\right)=0$
Answer
It is given that equation is ​​$\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\cos \left(\frac{d y}{d x}\right)=0$
We can see that the highest order derivative present in the given differential equation is $\frac{d^{2} y}{d x^{2}}$
Thus, its order is two. The given differential equation is not a polynomial equation in its derivative.
Therefore, its degree is not defined.
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Question 201 Mark
Determine order and degree (if defined) of differential equation ${\left( {\frac{{ds}}{{dt}}} \right)^4} + 3s\frac{{{d^2}s}}{{d{t^2}}} = 0$
Answer
Order = 2
Degree = 1
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Question 211 Mark
Determine order and degree (if defined) of differential equation y' + 5y = 0
Answer
It is given that equation is y’ + 5y = 0
We can see that the highest order derivative present in the differential is y’.
Thus, its order is one. It is polynomial equation in y’. The highest power raised to y’ is 1.
Therefore, its degree is one.
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Question 221 Mark
The order of the differential equation $2 x^{2} \frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+y=0$ is
Answer
It is given that equation is $2 x^{2} \frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+y=0$
We can see that the highest order derivative present in the given differential equation is $\frac{d^{2} y}{d x^{2}}$
Thus, its order is two. 
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Question 231 Mark
The degree of the differential equation $\left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\left(\frac{d y}{d x}\right)^{2}+\sin \left(\frac{d y}{d x}\right)+1=0$ is
Answer
It is given that equation is $\left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\left(\frac{d y}{d x}\right)^{2}+\sin \left(\frac{d y}{d x}\right)+1=0$
The given differential equation is not a polynomial equation in its derivative
Therefore, its degree is not defined.
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Question 241 Mark
Determine order and degree (if defined) of differential equation: y" + 2y' + siny = 0
Answer
It is given that equation is y’’ + 2y’ + siny = 0
We can see that the highest order derivative present in the differential is y”.
Thus, its order is two. It is polynomial equation in y” and y’
So, the highest power raised to y” is 1
Therefore, its degree is one.
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Question 251 Mark
Determine order and degree (if defined) of differential equation: $\frac{{{d^4}y}}{{d{x^4}}} + \sin \left( {y'''} \right)$ = 0
Answer
order = 4, degree = not defined
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Question 261 Mark
Find the order and degree, if defined, of the differential equation $y^{\prime \prime \prime}+y^{2}+e^{y^{\prime}}=0$
Answer
The highest order derivative present in the differential equation is y′′′, so its order is three. The given differential equation is not a polynomial equation in its derivatives and so its degree is not defined.
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Question 271 Mark
Find the order and degree, if defined, of the differential equation $x y \frac{d^{2} y}{d x^{2}}+x\left(\frac{d y}{d x}\right)^{2}-y \frac{d y}{d x}=0$
Answer
The highest order derivative present in the given differential equation is $\frac{d^{2} y}{d x^{2}}$, so its order is two. It is a polynomial equation in $\frac{d^{2} y}{d x^{2}}$ and $\frac{d y}{d x}$ and the highest power raised to $\frac{d^{2} y}{d x^{2}}$ is one, so its degree is one.
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Question 281 Mark
Find the order and degree, if defined, of the differential equation $\frac{d y}{d x}-\cos x=0$
Answer
The highest order derivative present in the differential equation is $\frac{d y}{d x}$, so its order is one. It is a polynomial equation in $\frac{d y}{d x}$ and the highest power raised to $\frac{d y}{d x}$ is one, so its degree is one.  
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