# Solve the differential equation #dy/dt = 4 sqrt(yt)# y(1)=6?

##### 2 Answers

#### Explanation:

We should separate the variables here by treating

#dy/dt=4sqrtysqrtt#

#dy/sqrty=4sqrttdt#

Integrating both sides and rewriting with fractional exponents:

#inty^(-1/2)dy=4intt^(1/2)dt#

Using typical integration rules:

#y^(1/2)/(1/2)=4(t^(3/2)/(3/2))+C#

#2sqrty=8/3t^(3/2)+C#

Solving for

#y=(4/3t^(3/2)+C)^2#

We were given the initial condition

#6=(4/3(1)^(3/2)+C)^2#

#sqrt6=4/3+C#

#C=sqrt6-4/3#

Then:

#y=(4/3t^(3/2)+sqrt6-4/3)^2#

and:

#### Explanation:

This is separable.

Differentiate both sides wrt t:

Chain rules allows us to re-write first term:

Then integrate:

Apply the IV:

So:

And: