Question

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

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

oops, yes, thats dy/dt

Answer 6

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

Answer 7

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

Answer 8

thanks !

Answer 9

sure:)