Question Number 227222 by Spillover last updated on 06/Jan/26
$$\:\:\:\:{Solve} \\ $$$$\:\:\:\:\:\:\:{x}\frac{{dy}}{{dx}}+{y}={x}^{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\mathrm{6}/\mathrm{1}/\mathrm{2026} \\ $$$$ \\ $$
Answered by som(math1967) last updated on 07/Jan/26
$${x}\frac{{dy}}{{dx}}\:+{y}={x}^{\mathrm{3}} \\ $$$$\Rightarrow\frac{{dy}}{{dx}}\:+\frac{{y}}{{x}}={x}^{\mathrm{2}} \\ $$$$\:{IF}={e}^{\int\frac{{dx}}{{x}}} ={e}^{{lnx}} ={x} \\ $$$$\Rightarrow{x}\frac{{dy}}{{dx}}\:+{y}={x}^{\mathrm{3}} \\ $$$$\Rightarrow\int{d}\left({xy}\right)=\int{x}^{\mathrm{3}} {dx} \\ $$$$\:\therefore{xy}=\frac{{x}^{\mathrm{4}} }{\mathrm{4}}+{c} \\ $$
Commented by Spillover last updated on 07/Jan/26
$${thanks} \\ $$
Answered by mr W last updated on 07/Jan/26
$$\frac{{d}}{{dx}}\left({xy}\right)={x}^{\mathrm{3}} \\ $$$${xy}=\frac{{x}^{\mathrm{4}} }{\mathrm{4}}+{C} \\ $$$$\Rightarrow{y}=\frac{\mathrm{1}}{{x}}\left(\frac{{x}^{\mathrm{4}} }{\mathrm{4}}+{C}\right) \\ $$
Commented by Spillover last updated on 07/Jan/26
$${thanks} \\ $$