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

Question

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

✓

Answer

Given differential equation is $\frac{\text{dy}}{\text{dx}}-\text{y}=\sin\text{x}.$

⇒ 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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