Question

y’+y=xsqrty решите задачу

Answer 1

$y'+y=x\sqrt{y}\\ \dfrac{y'}{\sqrt{y}}+\sqrt{y}=x\\ 2*\dfrac{y'}{2\sqrt{y}}*e^\frac{x}{2}+2*\sqrt{y}*\dfrac{1}{2}e^\frac{x}{2}=xe^\frac{x}{2}\\ \dfrac{y'}{2\sqrt{y}}*e^\frac{x}{2}+\sqrt{y}*\dfrac{1}{2}e^\frac{x}{2}=\dfrac{x}{2}e^\frac{x}{2}\\ (\sqrt{y}*e^\frac{x}{2})'_x=\dfrac{x}{2}e^\frac{x}{2}\\ \sqrt{y}*e^\frac{x}{2}=\int \dfrac{x}{2}e^\frac{x}{2} dx$

$(*)\; \int \dfrac{x}{2}e^\frac{x}{2} dx=[\frac{x}{2}=t]=2\int te^t dt=e^t(t-1)+C=e^\frac{x}{2}(\frac{x}{2}-1)+C\;(*) \\ \sqrt{y}*e^\frac{x}{2}=e^\frac{x}{2}(\frac{x}{2}-1)+C\\ \sqrt{y}=(\frac{x}{2}-1)+\dfrac{C}{e^\frac{x}{2}}\\ y=\left((\frac{x}{2}-1)+\dfrac{C}{e^\frac{x}{2}}\right)^2$