Question

The solution to the differential equation $(x+1)\frac{dy}{dx}-y=e^{3x}(x+1{)}^2$ is

• A. $y=(x+1)e^{3x}+c$
• B. $3y=(x+1)+e^{3x}+c$
• C. $\frac{3y}{x+1}=e^{3x}+c$
• D. $y{e}^{-3x}=3(x+1)+c$

Answer 1

The correct option is C $\frac{3y}{x+1}=e^{3x}+c$

The given equation is $\frac{dy}{dx}-\frac{y}{x+1}=e^{3x}(x+1)$
I.F. $=e^{\int -\frac{1}{x+1}dx}=e^{-log(x+1)}=\frac{1}{x+1}$
The solution is
$y\left(\frac{1}{x+1}\right)=\int e^{3x}(x+1).\frac{1}{x+1}dx+a$
$\Rightarrow \frac{y}{x+1}=\int e^{3x}dx+a=\frac{e^{3x}}{3}+a$
$\Rightarrow \frac{3y}{x+1}=e^{3x}+c,c=3a$