Solve the separable differential equation 8x6yx2105dydx 0 Su

Solve the separable differential equation

8x?6y(x^2+1)^(0.5)dydx =0.

Subject to the initial condition: y(0)=5 .
y=

Solution

8x?6y(x^2+1)^(0.5)dy/dx =0.

by variable separare

8xdx=6y(x^2+1)^(0.5)dy

8xdx/(x^2+1)^(0.5)=6ydy

putting z=(x^2+1)

dz=2xdx

so

4dz/z^0.5=6ydy

integrating both sides

4z^0.5/0.5=6y²/2+c

8z^0.5=3y²+c

8(x^2+1)^0.5=3y²+c

As  y(0)=5

so 8=75+c

c=-67

therefore

8(x^2+1)^0.5=3y²-67

y=(1/sqrt3)(64(x^2+1)²+67)^1/4