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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations4 Marks
Question
Solve the following differential equation:

$\frac{\text{dy}}{\text{dx}}+\text{y}=\text{e}^{-2\text{x}}$

✓

Answer

Here, $\frac{\text{dy}}{\text{dx}}+\text{y}=\text{e}^{-2\text{x}}$

This is a linear differential equation, comparing it with

$\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$

$\text{P}=1,\text{Q}=\text{e}^{-2\text{x}}$

I.F. $=\text{e}^{\int\text{Pdx}}$

$=\text{e}^{\int2\text{dx}}$

$=\text{e}^{\text{x}}$

Solution of the equation is given by,

$\text{y}\times(\text{I.F.})=\int\text{Q}\times(\text{I.F.})\text{dx + C}$

$\text{y}\times\text{e}^{\text{x}}=\int\text{e}^{-2\text{x}}\times\text{e}^{\text{x}}\text{dx + C}$

$=\int\text{e}^{-\text{x}}+\text{C}$

$\text{y}\text{e}^{\text{x}}=\frac{\text{e}^{-\text{x}}}{-1}+\text{C}$

$\text{y}=-\text{e}^{-2\text{x}}+\text{C}\text{e}^{-\text{x}}$

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