Solve the differential equation dy dx +1=e^ x+y . - Vidyadip

Question

Solve the differential equation $\frac{\text{dy}}{\text{dx}}+1=\text{e}^{\text{x+y}}.$

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Answer

We have $\frac{\text{dy}}{\text{dx}}+1=\text{e}^{\text{x+y}}\ ....(\text{i})$
Take $\text{x}+\text{y}=\text{t}$
$\Rightarrow1+\frac{\text{dy}}{\text{dx}}=\frac{\text{dt}}{\text{dx}}$
Substituting $\text{x}+\text{y}=\text{t}$ in equation (i) we get,
$\frac{\text{dt}}{\text{dx}}=\text{e}^\text{t}$
$\Rightarrow\text{e}^{-\text{t}\text{dt}}=\text{dx}$
$\Rightarrow-\text{e}^{-\text{t}}=\text{x}+\text{C}$
$\Rightarrow\frac{-1}{\text{e}^\text{x+y}}=\text{x}+\text{C}$
$\Rightarrow-1=(\text{x}+\text{C})\text{e}^\text{x+y}$
$\Rightarrow(\text{x}+\text{C})\text{ e}^\text{x+y}1=0$

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