kasibug1v

2022-09-29

How do you find the particular solution to $\sqrt{x}+\sqrt{y}y\prime =0$ that satisfies y(1)=4?

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Demarion Thornton

Expert

$\sqrt{x}+\sqrt{y}y\prime =0$
This is separable:

$\frac{2}{3}{y}^{\frac{3}{2}}=-\left(\frac{2}{3}{x}^{\frac{3}{2}}+C\right)$
$⇒{y}^{\frac{3}{2}}=-{x}^{\frac{3}{2}}+C$
$y\left(1\right)=4⇒{\left(y\left(1\right)\right)}^{\frac{3}{2}}=8=-1+C⇒C=9$
$⇒y={\left(9-{x}^{\frac{3}{2}}\right)}^{\frac{2}{3}}$

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