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Differential Equation and Applications (p-1) — Maths (commerce) STD 12 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Differential Equation and Applications (p-1)3 Marks
Question
Solve the following differential equations : $\frac{d y}{d x}+2 x y=x$
✓

Answer

$
\frac{d y}{d x}+2 x y= x
$
This is the linear differential equation of the form
$
\begin{gathered}
\frac{d y}{d x}+P \cdot y=Q, \text { where } P=2 x, Q=x \\
\therefore \text { I.F. }=e^{\int P d x}=e^{\int 2 x d x} \\
=e^{2 \int x d x}=e^{2\left(\frac{x^2}{2}\right)}=e^{x^2}
\end{gathered}
$
$\therefore$ the solution of $(1)$ is given by
$
\begin{aligned}
& y \cdot\left(\text { I.F.) }=\int Q \cdot \text { (I.F.) } d x+c\right. \\
& \therefore y e^{x^2}=\int x e^{x^2} d x+c
\end{aligned}
$
Put $x^2=t \quad \therefore 2 x d x=d t$
$
\therefore x d x=\frac{1}{2} d t
$
$\therefore$ (1) becomes
$
\begin{aligned}
& y e^{x^2}=\frac{1}{2} \int e^t d t+c \\
& \therefore y e^{x^2}=\frac{1}{2} e^{x^2}+c
\end{aligned}
$
This is the general solution.

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