Find general solution to the equation $xy^{'}=2x^2\sqrt{y}+4y$ This is Bernoulli's differential equation:
$$xy^{'}-4y=2x^2y^{\frac{1}{2}}$$
Substitution $y=z^{\frac{1}{1-\alpha}},\alpha=\frac{1}{2},y^{'}=z^{'2}$ gives $$xz^{'2}-4z^2=2x^2z$$
Is this correct? What is the method for solving this equation?
 A: $$xy'(x)=2x^2\sqrt{y(x)}+4y(x)\Longleftrightarrow$$
$$xy'(x)-4y(x)=2x^2\sqrt{y(x)}-4y(x)\Longleftrightarrow$$
$$\frac{y'(x)}{2\sqrt{y(x)}}-\frac{2\sqrt{y(x)}}{x}=x\Longleftrightarrow$$

Let $v(x)=\sqrt{y(x)}$; which gives $v'(x)=\frac{y'(x)}{2\sqrt{y(x)}}$:

$$v'(x)-\frac{2v(x)}{x}=x\Longleftrightarrow$$

Let $\mu(x)=e^{\int-\frac{2}{x}\space\text{d}x}=\frac{1}{x^2}$. 
Multiply both sides by $\mu(x)$:

$$\frac{v'(x)}{x^2}-\frac{2v(x)}{x^3}=\frac{1}{x}\Longleftrightarrow$$

Substitute $-\frac{2}{x^3}=\frac{\text{d}}{\text{d}x}\left(\frac{1}{x^2}\right)$:

$$\frac{v'(x)}{x^2}+\frac{\text{d}}{\text{d}x}\left(\frac{1}{x^2}\right)v(x)=\frac{1}{x}\Longleftrightarrow$$

Apply the reverse product rule $g\frac{\text{d}f}{\text{d}x}+f\frac{\text{d}g}{\text{d}x}=\frac{\text{d}}{\text{d}x}(fg)$ to the left-hand side:

$$\frac{\text{d}}{\text{d}x}\left(\frac{v(x)}{x^2}\right)=\frac{1}{x}\Longleftrightarrow$$
$$\int\frac{\text{d}}{\text{d}x}\left(\frac{v(x)}{x^2}\right)\space\text{d}x=\int\frac{1}{x}\space\text{d}x\Longleftrightarrow$$
$$\frac{v(x)}{x^2}=\ln\left|x\right|+\text{C}\Longleftrightarrow$$
$$v(x)=x^2\left(\ln\left|x\right|+\text{C}\right)\Longleftrightarrow$$
$$y(x)=\left(x^2\left(\ln\left|x\right|+\text{C}\right)\right)^2\Longleftrightarrow$$
$$y(x)=x^4\left(\ln\left|x\right|+\text{C}\right)^2$$
A: We are given
$$xy^{'}-4y=2x^2y^{\frac{1}{2}}$$
Let $z = y^{1-\frac{1}{2}} = y^{\frac 12}.$ Then
$$z' = \frac12 y^{-\frac12}y' \implies y' = 2\frac{z'}{y^{-\frac12}}.$$With this, we get
$$2x\frac{z'}{y^{-\frac12}}-4y=2x^2y^{\frac{1}{2}}$$
Now multiply by $y^{-\frac12}$. We get
$$2x{z'}-4y^{\frac12}=2x^2$$ which is the same as
$$2x{z'}-4z=2x^2$$
Now solve this last equation (linear first order).
A: If y = z^2 ... when you differentiate this equation with respect to x ... notice z is implicitly dependant on x too. So take this into account.
Hint: Chain rule
