Find the general solution of the differential equation dy dx -y= … - Vidyadip

Question

Find the general solution of the differential equation $\frac{\text{dy}}{\text{dx}}-\text{y}=\sin\text{x}.$

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Answer

Given differential equation is $\frac{\text{dy}}{\text{dx}}-\text{y}=\sin\text{x}.$
$\Rightarrow$ Integrating factor $= \text{e}^{-\text{x}}$
$\therefore\ $ Solution is: $\lambda e^{–x} = \int\sin \text{e}^{-\text{x}}\text{dx}=\text{I}_1$
$\text{I}_1=-\sin \text{x}\text{e}^{-\text{x}}+\int\cos \text{x}\text{e}^{-\text{x}}\text{dx}$$=-\sin \text{x}\text{e}^{-\text{x}}+[-\cos \text{x}\text{e}^{-\text{x}}-\int+\sin\text{x}\text{e}^{-\text{x}}\text{dx}]$
$\text{I}_1=\frac{1}{2}[-\sin\text{x}-\cos\text{x}]\text{e}^{-\text{x}}$
$\therefore\ $ Solution is: $\lambda e^{–x }$ $=\frac{1}{2}(-\sin\text{x}-\cos\text{x})\text{e}^{-\text{x}}+\text{c} \text{or}\ \text{y}=-\frac{1}{2}(\sin\text{x}+\cos\text{x})+\text{ce}^\text{x}$

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