Given the equation:
$$\frac{d}{d x} y{\left(x \right)} = x \sqrt{y{\left(x \right)}}$$
This differential equation has the form:
f1(x)*g1(y)*y' = f2(x)*g2(y),
where
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = x$$
$$\operatorname{g_{2}}{\left(y \right)} = \sqrt{y{\left(x \right)}}$$
We give the equation to the form:
g1(y)/g2(y)*y'= f2(x)/f1(x).
Divide both parts of the equation by g2(y)
$$\sqrt{y{\left(x \right)}}$$
we get
$$\frac{\frac{d}{d x} y{\left(x \right)}}{\sqrt{y{\left(x \right)}}} = x$$
We separated the variables x and y.
Now, multiply the both equation sides by dx,
then the equation will be the
$$\frac{dx \frac{d}{d x} y{\left(x \right)}}{\sqrt{y{\left(x \right)}}} = dx x$$
or
$$\frac{dy}{\sqrt{y{\left(x \right)}}} = dx x$$
Take the integrals from the both equation sides:
- the integral of the left side by y,
- the integral of the right side by x.
$$\int \frac{1}{\sqrt{y}}\, dy = \int x\, dx$$
Detailed solution of the integral with yDetailed solution of the integral with xTake this integrals
$$2 \sqrt{y} = Const + \frac{x^{2}}{2}$$
Detailed solution of the equationWe get the simple equation with the unknown variable y.
(Const - it is a constant)
Solution is:
$$\operatorname{y_{1}} = y{\left(x \right)} = \frac{C_{1}^{2}}{4} + \frac{C_{1} x^{2}}{4} + \frac{x^{4}}{16}$$