Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS3 Marks
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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