Question

Assertion :The general solution of the differential equation $x\frac{dy}{dx}+2y=x^2(x\ne 0)$ is $y=\frac{x^2}{4}+k{x}^{-2}$ Reason: The number of arbitrary constant in the general solution is equal to the order of the differential equation.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Given differential equation is
$x\frac{dy}{dx}+2y=x^2$
$\Rightarrow$ $\frac{dy}{dx}+\frac{2}{x}y=x$
I.F.$=e^{\int \frac{2}{x}dx}=e^{\log x^2}$

$\therefore$ Solution is given by
$y(I.F.)=\int x(I.F.)dx$, $y{x}^2=\int x^{3}dx$
$y=\frac{x^4}{4}+c{x}^{-2}$ (general solution)
$\therefore$ Assertion (A) is true

Again, order of the differential equation is equal to the number of arbitrary constant is true but it is not the correct explanation of Assertion (A).