Solution to a specific form of Differential Equation

Question

In my textbooks, I can't seem to find a solution for the differential equation of the form: $$y' = 3t*\sqrt y$$ How do I solve this differential equation? Can I classify this equation as a Bernoulli Differential Equation, with $n=1/2$?

Answer 1

From

$$\frac{y'}{2\sqrt y}=\frac32t,$$

you draw

$$\sqrt y=\frac34t^2+C,$$

$$y=\left(\frac34t^2+C\right)^2.$$

Note that $y=0$ is also a solution.

Also note that a square root is positive, so that $\dfrac34t^2+C>0$ must hold. When $C<0$ this limits the domain.

Answer 2

Why would you need to? The equation is seperable $$ \frac{y'}{\sqrt{y}} = 3t $$

which integrates to $$ \sqrt{y} = \frac{3t^2}{4} + c $$