## S 2 years ago use separation of variables to solve differential equation :dy/dx=t/(y+1)^(1/2) , y(1)=3 ?

1. .Sam.

multiply both sides by$\sqrt{y+1}$ Then integrate $\frac{dy}{dx} \sqrt{y+1}\text{ = }t$ $\sqrt{y+1}dy=tdx$ $\int\limits_{}^{}\sqrt{y+1}dy=\int\limits_{}^{} tdx$

2. S

I got this part, but I am confused how to solve it

3. .Sam.

$\int\limits \sqrt{y+1} \, dy=\frac{2}{3} (y+1)^{3/2}$ $\int\limits_{}^{}tdx=tx+c$ ------------------------------------------------ $\frac{2}{3} (y+1)^{3/2} =tx+c$ $(y+1)^{3/2}=\frac{3(tx+c)}{2}$ $y+1=(\frac{3(tx+c)}{2})^{2/3}$ $y=(\frac{3(tx+c)}{2})^{2/3}-1$

4. Algebraic!

I think it's supposed to be dy/dt ?

5. S

oops, yes, thats dy/dt

6. Algebraic!

then it's 2/3*(y+1)^(3/2) = (t^2)/2 +C

7. Algebraic!

use I.V. to find C , solve for y(t)

8. S

thanks !

9. Algebraic!

sure:)