1 Marks Question

Question 11 Mark

How many number of arbitrary constants in the general solution of a differential equation of fourth order.

Answer

Three will be 4 arbitrary constants in the general solution of a differential equation of fourth order.

Question 21 Mark

Write the integrating factor of the equation $\frac{d y}{d x}+\frac{1}{\sin x}$ $y=e^x$.

Answer

$
\begin{aligned}
\text { I.F. } & =e^{\int Pdx}=e^{\int \frac{1}{\sin x} d x} \\
\text { I.F. } & =e^{\int \operatorname{cosec} x d x}=e^{\log \tan \frac{x}{2}} \\
& =\tan \frac{x}{2}
\end{aligned}
$

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Question 31 Mark

Write the general solution of the equation $\frac{d y}{d x}+\frac{1}{x} y=$ $x$.

Answer

$P =\frac{1}{x}$
$\therefore$
$
\begin{aligned}
\text { I.F. } & =e^{\int P d x}=e^{\int \frac{1}{x} d x} \\
\text { I.F. } & =e^{\log x}=x \\
y . \text { I.F. } & =\int(\text { I.F. } \times \text { Q }) d x \\
y \times x & =\int x \times x d x=\int x^2 d x
\end{aligned}
$
$
\Rightarrow \quad x y=\frac{x^3}{3}+c
$

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Question 41 Mark

Solve : $\frac{d y}{d x}=5 x+7$

Answer

$
\begin{array}{ll}
\frac{d y}{d x}=5 x+7 & \\
\quad \therefore d y=(5 x+7) d x
\end{array}
$
On integrating $\int d y=\int(5 x+7)=\frac{5 x^2}{2}+7 x+ C$
$\therefore y=\frac{5}{2} x^2+7 x+ C$, where C is an integration constant.

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Question 51 Mark

What will be the differential equation of the curve $y=$ $A e^x+ B e^{-x}$ ?

Answer

$y= A e^x+ B e^{-x}$
$
\begin{aligned}
\frac{d y}{d x} & =A e^x-B e^{-x} \\
\therefore \quad \frac{d^2 y}{d x^2} & =A e^x+B e^{-x}=y \\
\frac{d^2 y}{d x^2} & =y
\end{aligned}
$

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Question 61 Mark

What will be the order and degree of the differential equation $x y \frac{d y}{d x}=\left(\frac{1+y^2}{1+x^2}\right)\left(1+x+x^2\right)$ ?

Answer

I. order $=1$, degree $=1$.

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Question 71 Mark

What will be the order and degree of the differential equation $\frac{d^4 y}{d x^4}-4 \frac{d y}{d x}+4 y=5 \cos 3 x ?$

Answer

Order $=4$, degree $=1$

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Question 81 Mark

What is the form of the differential equation $\cos (x+y)$
$
\frac{d y}{d x}=1 ?
$

Answer

Equation reducible to variables separable.

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Question 91 Mark

What is the form of the differential equation $\frac{d y}{d x}-y \tan$ $x=e^x \sec x ?$

Answer

This is a linear equation because the form of this equation is $\frac{d y}{d x}+ P y= Q$.

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