Find the particular solution, y=f(x), to the differential equation (1+x^2 ) dy/dx=x^3/y where f(0)=-3.
(1+x^2 ) dy/dx = x^3/y
Cross-multiplying,
y dy/dx = x^3/(1 + x^2)
y dy = x^3/(1 + x^2) dx
int y dy = int x^3/(1 + x^2) dx = int (x + x^3)/(1 + x^2) - x/(1 + x^2) dx
int y dy = int x dx – int x/(1 + x^2) dx
y^2/2 = x^2/2 – (1/2)ln(1 + x^2) + C
y^2 = x^2 – ln(1 + x^2) + D
y = √(x^2 – ln(1 + x^2) + D)
Using the initial condition, f(0) = -3,
-3 = √(0 - ln(1) + D)
D = 9
The particular solution is.
f(x) = √( x^2 – ln(1 + x^2) + 9)