For each of the differential equations given in find the general … - Vidyadip

Question

For each of the differential equations given in find the general solution:
$\frac{\text{dy}}{\text{dx}}+3\text{y}=\text{e}^{-2\text{x}}$

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Answer

The given differential equation is $\frac{\text{dy}}{\text{dx}}+\text{py}=\text{Q} ($where $p = 3$ and $Q = e ^{-2x})$
$\text{Now, I.F}=\text{e}^{\int\text{pdx}}=\text{e}^{\int3\text{dx}}=\text{e}^{3\text{x}}.$
The solution of the given differential equation is given by the relation,
$\text{y}(\text{I.F})=\int(\text{Q}\times\text{I.F.})\text{dx}+\text{C}$
$\Rightarrow\ \text{ye}^{\text{3}\text{x}}=\int(\text{e}^{-2\text{x}}\times\text{e}^{3\text{x}})+\text{C}$
$\Rightarrow\ \text{ye}^{\text{3}\text{x}}=\int\text{e}^\text{x}\text{dx}+\text{C}$
$\Rightarrow\ \text{ye}^{\text{3}\text{x}}=\text{e}^\text{x}+\text{C}$
$\Rightarrow\ \text{y}=\text{e}^{\text{-2}\text{x}}+\text{C e}^{\text{-3}\text{x}}$
This is the required general solution of the given differential equation.

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