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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)2 Marks
Question
Solve the following differential equations : $\frac{d y}{d x}+y=e^{-x}$
✓

Answer

$
\frac{d y}{d x}+y=e^{-x}
$
This is the linear differential equation of the form
$
\begin{aligned}
& \frac{d y}{d x}+P \cdot y=Q, \text { where } P=1 \text { and } Q=e^{-x} \\
& \therefore \text { I.F. }=e^{\int P d x}=e^{\int 1 d x}=e^x
\end{aligned}
$
$\therefore$ 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 \cdot e^x=\int e^{-x} \cdot e^x d x+c \\
\therefore & e^x \cdot y=\int 1 d x+c \\
\therefore & e^x \cdot y=x+c \quad \therefore y e^x=x+c
\end{aligned}
$
This is the general solution.

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