M.C.Q (1 Marks)

Question 511 Mark

Let the differential equation is $\left[1+\left(\frac{d y}{d x}\right)^2\right]^{\frac{3}{2}}=\frac{d^2 y}{d x^2}$. Which of the following statements is/are true?
(i) Degree of the differential equation is 2 .
(ii) Order of the differential equation is 3 .
(iii) Order and degree of differential equation respectively are 2,2 .

Answer

(d) : The given differential equation can be written as
$
\left[1+\left(\frac{d y}{d x}\right)^2\right]^3=\left(\frac{d^2 y}{d x^2}\right)^2
$
Clearly, order and degree of given differential equation are 2,2 respectively.

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Question 521 Mark

The differential equation having solution as $y=17 e^x+a e^{-x}$ is

Answer

(b): We have, $y=17 e^x+a e^{-x} \Rightarrow y^{\prime}=17 e^x-a e^{-x}$
$
\Rightarrow y^{\prime \prime}=17 e^x+a e^{-x} \Rightarrow y^{\prime \prime}=y \Rightarrow y^{\prime \prime}-y=0
$

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Question 531 Mark

Integrating factor of the differential equation $\left(1-x^2\right) \frac{d y}{d x}-x y=1$ is

Answer

(c): $\left(1-x^2\right) \frac{d y}{d x}-x y=1 \Rightarrow \frac{d y}{d x}-\frac{x}{1-x^2} \cdot y=\frac{1}{1-x^2}$
$\therefore \quad$ I.F. $=e^{-\int \frac{x}{1-x^2} d x}=e^{\frac{1}{2} \int \frac{-2 x}{1-x^2} d x}$
$=e^{\frac{1}{2} \log \left(1-x^2\right)}=e^{\log \left(1-x^2\right)^{\frac{1}{2}}}=\sqrt{1-x^2}$

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Question 541 Mark

The integrating factor of the differential equation $\left(x+3 y^2\right) \frac{d y}{d x}=y$ is

Answer

(c) : We have, $\left(x+3 y^2\right) \frac{d y}{d x}=y$
$
\Rightarrow \frac{x+3 y^2}{y}=\frac{d x}{d y} \Rightarrow \frac{d x}{d y}-\frac{x}{y}=3 y
$
This is a linear differential equation.
$
\therefore \quad \text { I.F. }=e^{-\int \frac{d y}{y}}=e^{-\log y}=e^{\log y^{-1}}=\frac{1}{y}
$

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Question 551 Mark

Order and degree of the differential equation $\left(1+\left(\frac{d y}{d x}\right)^3\right)^{\frac{7}{3}}=7 \frac{d^2 y}{d x^2}$ are respectively

Answer

(b) : We have $\left(1+\left(\frac{d y}{d x}\right)^3\right)^{\frac{7}{3}}=7 \frac{d^2 y}{d x^2}$
$
\Rightarrow\left(1+\left(\frac{d y}{d x}\right)^3\right)^7=\left(7 \frac{d^2 y}{d x^2}\right)^3
$
$\therefore \quad$ Order is 2 and degree is 3 .

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Question 561 Mark

If $\log _e\left(1+\frac{d^2 y}{d x^2}\right)=x$, then find the sum of order and degree of given differential equation.

Answer

(c) : Given differential equation is $\log _e\left(1+\frac{d^2 y}{d x^2}\right)=x$
$\Rightarrow 1+\frac{d^2 y}{d x^2}=e^x$
Here, highest order derivative is $\frac{d^2 y}{d x^2}$, whose power is 1
$\therefore \quad$ Its order is 2 and degree is 1 .
$\therefore \quad$ Required sum $=2+1=3$.

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Question 571 Mark

If $\frac{d y}{d x}=y \sin 2 x, y(0)=1$, then solution is

Answer

(a) : We have, $\frac{d y}{d x}=y \sin 2 x$
$
\Rightarrow \quad \frac{d y}{y}=\sin 2 x d x \Rightarrow \log y=-\frac{\cos 2 x}{2}+C
$
Since $y(0)=1 \Rightarrow x=0, y=1 \Rightarrow C=1 / 2$
$
\therefore \quad \log y=\frac{1}{2}(1-\cos 2 x) \Rightarrow \log y=\sin ^2 x \Rightarrow y=e^{\sin ^2 x}
$

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Question 581 Mark

The solution of the differential equation $\frac{d y}{d x}=\frac{x^2+1}{2 x y}$ satisfying $y(1)=1$, is

Answer

(a) : Given, $\frac{d y}{d x}=\frac{x^2+1}{2 x y}$
$
\Rightarrow 2 y d y=\frac{\left(x^2+1\right)}{x} d x \Rightarrow 2 y d y=\frac{(x+1)}{x} d x
$
Integrating on both sides, we get
$
\begin{aligned}
& y^2=\frac{x^2}{2}+\ln |x|+C \\
\Rightarrow \quad & y^2=\frac{x^2}{2}+C
\end{aligned}
$
When $x=1, y=1$
$
\therefore \quad C=\frac{1}{2} \quad \therefore \quad y^2=\frac{x^2}{2}+\ln |x|+\frac{1}{2}
$

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Question 591 Mark

The general solution of the differential equation $x\left(1-y^2\right) d x+y\left(1-x^2\right) d y=0$ is

Answer

(b) : Given differential equation is
$
\begin{aligned}
x\left(1-y^2\right) d x+y\left(1-x^2\right) d y & =0 \\
\Rightarrow\left(\frac{x}{1-x^2}\right) d x+\left(\frac{y}{1-y^2}\right) d y & =0
\end{aligned}
$
On integrating, we get
$
\begin{aligned}
& \frac{1}{2} \log \left(1-x^2\right)-\frac{1}{2} \log \left(1-y^2\right)=k \\
\Rightarrow \quad & \log \left(1-x^2\right)\left(1-y^2\right)=-2 k \Rightarrow\left(1-x^2\right)\left(1-y^2\right)=e^{-2 k}=C
\end{aligned}
$

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MCQ 601 Mark

Integrating factor of differential question $\frac{d y}{d x}-y=\cos x$ is-

A
$\sin x$
B
$\cos x$
C
$e^x$
✓
$e^{-x}$

Answer

Correct option: D.

$e^{-x}$

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MATHS M.C.Q (1 Marks)