If y = y ( x ) is the solution of the differential equation, x d y d x + 2 y = x 2 satisfying y 1 = 1 , then y 1 2 is equal to

Question

Aicharya · Accepted Answer

Given differential equation can be written as

\frac{d y}{d x} + \frac{2 y}{x} = x

, which is a linear differential equation.

If

= e^{\int \frac{2}{x} d x} = e^{2 l n x} = x^{2}

Solution is

y . x^{2} = \int x . x^{2} d x

\Rightarrow y . x^{2} = \frac{x^{4}}{4} + c

Since, given curve passes through

(1, 1)

\Rightarrow c = \frac{3}{4}

Hence,

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

.

So,

y (\frac{1}{2}) = \frac{49}{16}

.

If y = y ( x ) is the solution of the differential equation, x d y d x + 2 y = x 2 satisfying y 1 = 1 , then…