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If $y = y(x)$ is the solution of the differential equation, $x\frac{dy}{dx} +2y = x^2$ satisfying $y (1) =1$ then $y\left(\frac{1}{2} \right )$ is equal to
Option 1)
$\frac{7}{64}$
Option 2)
$\frac{1}{4}$
Option 3)
$\frac{49}{16}$
Option 4)
$\frac{13}{16}$

Answers (1)

best_answer

Linear Differential Equation -

$\frac{dy}{dx}+Py= Q$

- wherein

P, Q are functions of x alone.

Linear Differential Equation -

Multiply by $e^{SPdx}$ which is the Integrating factor

- wherein

P is the function of x alone

Differential Equation can be written as.

$\frac{\mathrm{d} y}{\mathrm{d} x}+2\left ( \frac{y}{x} \right )=x$

from the concept we have learnt

$\frac{\mathrm{d} y}{\mathrm{d} x}+P(x)\cdot y=Q(x)$

If = $e^{\int P\cdot dx}=x^{2}$

Solution of this d.e is

$Y(IF)=\int Q\left ( IF \right )dx+C$

So, $\therefore yx^{2}=\frac{x^{4}}{4}+C$

given $y(1)=1$

$1=\frac{1}{4}+C\Rightarrow C=\frac{3}{4}$

$y\left ( x^{2} \right )=\frac{x^{4}}{4}+\frac{3}{4}$

We need to find $y\left ( x=\frac{1}{2} \right )$

Put $x=\frac{1}{2}$

$\frac{y}{4}=\frac{1}{64}+\frac{3}{4}$

$y=\frac{49}{16}$

Option 1)

$\frac{7}{64}$

Option 2)

$\frac{1}{4}$

Option 3)
$\frac{49}{16}$
Option 4)

$\frac{13}{16}$

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