Q54 / 100 · BITSAT 2024 (Memory Based Paper 3) · 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
Given, $\frac{d y}{d x}-\frac{y}{x+1}=e^{3 x}(x+1)$
Above is linear differential equation of form
$\begin{aligned}
& \frac{d y}{d x}+P x=Q \\
& \therefore \quad \mathrm{IF}=e^{\int P d x}=e^{\int-\frac{1}{x+1} d x}=e^{-\ln (1+x)}=\frac{1}{1+x}
\end{aligned}$
$\Rightarrow$ Solution will be
$\begin{aligned}
y \cdot \mathrm{IF} & =\int Q \cdot \mathrm{IF} d x \\
y \cdot \frac{1}{1+x} & =\int e^{3 x}(x+1) \times \frac{1}{(1+x)} d x \\
\frac{y}{1+x} & =\frac{e^{3 x}}{3}+C^{\prime} \\
\frac{3 y}{1+x} & =e^{3 x}+C^{\prime} \quad \quad\left[\because C=3 C^{\prime}\right]
\end{aligned}$
Hence, the correct answer is option: 'C'.