The solution of the equation d y d x +2 y=4 x is :

Question

The solution of the equation $\frac{d y}{d x}+2 y=4 x$ is :

✓

Answer

(A)
$
\frac{d y}{d x}+2 y=4 x
$
Here $P =2$ and $Q =4 x$
$
\text { I.F. }=e^{\int P d x}=e^{\int 2 d x}=e^{2 x}
$
Required solution is
$
\begin{aligned}
& & y \times e^{2 x} & =\int e^{2 x} \cdot 4 x d x+c \\
\Rightarrow & & y \cdot e^{2 x} & =4 \int x e^{2 x} d x+c \\
\Rightarrow & & y \cdot e^{2 x} & =4\left(\frac{x \cdot e^{2 x}}{2}-\int \frac{1 \cdot e^{2 x}}{2} d x\right)+c \\
\Rightarrow & & y \cdot e^{2 x} & =2 x e^{2 x}-e^{2 x}+C \\
\Rightarrow & & y & =(2 x-1)+C e^{-2 x}
\end{aligned}
$
Hence the correct choice is (A).

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