Q57 / 125 · BITSAT 2020 · Mathematics
The solution of the differential equation
$(x+1) \frac{d y}{d x}-y=e^{3 x}(x+1)^{2}$ is
- A$y=(x+1) e^{3 x}+c$
- B$3 y=(x+1)+e^{3 x}+c$
- Correct answerC$\frac{3 y}{x+1}=e^{3 x}+c$
- D$y e^{-3 x}=3(x+1)+c$
Solution
The given equation is $\frac{d y}{d x}-\frac{y}{x+1}=e^{3 x}(x+1)$
I.F. $=e^{\int-\frac{1}{x+1} d x}=e^{-\log (x+1)}=\frac{1}{x+1}$
The solution is
$y\left(\frac{1}{x+1}\right)=\int e^{3 x}(x+1) \cdot \frac{1}{x+1} d x+a$
$\Rightarrow \frac{y}{x+1}=\int e^{3 x} d x+a=\frac{e^{3 x}}{3}+a$
$\Rightarrow \frac{3 y}{x+1}=e^{3 x}+c, c=3 a$
Hence, the correct answer is option: 'C'.