Question

Assertion (A): The general solution of $\frac{dy}{d_X}-\frac{2}{x+1}y=(x+1{)}^3$ is $\frac{2y}{(x+1{)}^2}=(x+1{)}^2+c$
Reason (R) : The general solution of D.E is y (I.F)= $\int Q(I.F)dx$

• A. A and R both are true and R is correct explanation of A
• B. A and R both are true and R is not the correct explanation of A
• C. Only A is true
• D. Only R is true

Answer 1

The correct option is A A and R both are true and R is correct explanation of A

$\frac{dy}{dx}-\frac{2}{x+1}y=(x+1{)}^3$

Hence

$IF=e^{\int \frac{-2}{x+1}.dx}$

$=e^{-2ln(x+1)}$

$=e^{-ln(x+1{)}^2}$

$=\frac{1}{(x+1{)}^2}$

Thus the above differential equation changes to

$\frac{1}{(x+1{)}^2}.\frac{dy}{dx}-\frac{2}{(x+1{)}^3}.y=(x+1)$

$\frac{d\left(\frac{y}{(x+1{)}^2}\right)}{dx}=(x+1)$

$\frac{y}{(x+1{)}^2}=\int (x+1).dx$

$\frac{y}{(x+1{)}^2}=\frac{x^2}{2}+x+c$

$\frac{2y}{(x+1{)}^2}=x^2+2x+c$

$\frac{2y}{(x+1{)}^2}=(x+1{)}^2+C$ where $C=c-1$