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Question Number 224 by 123456 last updated on 25/Jan/15
$$\mathrm{solve} \\ $$$${x}\frac{{dy}}{{dx}}+{y}=\alpha{x}+\beta \\ $$
Answered by mreddy last updated on 16/Dec/14
$$\frac{{dy}}{{dx}}+\frac{{y}}{{x}}=\frac{\alpha{x}+\beta}{{x}} \\ $$$$\mathrm{Integrating}\:\mathrm{Factor}={e}^{\int\frac{\mathrm{1}}{{x}}{dx}} ={e}^{\mathrm{ln}\:{x}} ={x} \\ $$$${xy}=\int\frac{\alpha{x}+\beta}{{x}}\centerdot{x}\:{dx}\:+\:{C} \\ $$$${xy}=\frac{\alpha{x}^{\mathrm{2}} }{\mathrm{2}}+\beta{x}+{C} \\ $$$${y}=\frac{\alpha{x}}{\mathrm{2}}+\beta+{C} \\ $$