Solving a nonlinear differential equation: $y’=-y+ty^{1/2}$ with $y(2)=2$. Which analytical method can I use to solve this differential equation?
$y’=-y+ty^{1/2}$
With $y(2)=2$
I tried some direct methods but it didn’t work.
 A: You can start by adding both sides of your ODE by $y$:
$$y'+y=ty^{1/2} \tag{1}$$

An ordinary differential equation in the form
  $$y'+p(t)y=q(t)y^n \tag{2}$$
  where $n\in \mathbb{R}\setminus \{0,1\}$ is called a Bernoulli Differential Equation.

This is the case for equation $(1)$. It is well known that the change of variable $v=y^{1-n}$ reduces all Bernoulli Differential Equations into a first-order linear one. I will demonstrate this below, in general. Let's first divide both sides of $(2)$ by $y^n$:
$$y^{-n}y'+p(t)y^{1-n}=q(t) \tag{3}$$
Let's apply the change of variable (This follows from the chain rule):
$$v=y^{1-n} \implies v'=(1-n)y^{-n}y'$$
Substituting the above gives a linear ODE, as required.
$$\frac{1}{1-n}v'+p(t)v=q(t) \tag{4}$$

A linear nonhomogeneous differential equation can easily be solved via the integrating factor method. Be sure to not forget to implement the condition $y(2)=2$ after finding the general solution to $(1)$.
A: write $$\frac{\frac{dy}{dt}}{2\sqrt{y(t)}}+\frac{\sqrt{y(t)}}{2}=\frac{t}{2}$$
and set $$v(t)=\sqrt{y(t)}$$ then you will get
$$\frac{dv(t)}{dt}+\frac{v(t)}{2}=\frac{t}{2}$$
can you finish?
finally you will get $$-t+2-{{\rm e}^{-t/2}}{\it \_C1}+\sqrt {y \left( t \right) }=0$$
