Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS3 Marks
Question
Find the general solution of $\frac{\text{dy}}{\text{dx}}+\text{ay}=\text{e}^{\text{mx}}.$
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Answer
We have, $\frac{\text{dy}}{\text{dx}}+\text{ay}=\text{e}^{\text{mx}}$
which is a linear differential equation.
On comparing it with $\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q},$ we get
$\text{P}=\text{a},\text{Q}=\text{e}^{\text{mx}}$
$\text{I.F.}=\text{e}^{\int\text{P}\text{dx}}$
$\text{I.F.}=\text{e}^{\int\text{a}\text{dx}}$
$\text{I.F.}=\text{e}^{\text{ax}}$
The general solution is,
$\text{y.}\text{e}^{\text{ax}}=\int\text{e}^{\text{mx}}.\text{e}^{\text{ax}}\text{dx}+\text{C}$
$\Rightarrow\text{y.}\text{e}^{\text{ax}}=\int\text{e}^{(\text{m}+\text{a})\text{x}}\text{ dx}+\text{C}$
$\Rightarrow\text{y}.\text{e}^\text{ax}=\frac{\text{e}^{(\text{m}+\text{a})\text{x}}}{(\text{m}+\text{a})}+\text{C}$
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