Find the general solution of the following differential equations.

Question

Find the general solution of the following differential equations.

1. dy/dx= -x+1/2(y+3)

2. dy/dx= y²(1+ e^x )

Answer 1

Rewrite the equation as dy/dx -y/2 =- x +3/2. use an integration factor I= e^-∫dx/2 which is now I= e^-x/2. So the equation is now d(ye^-x/2) = -xe^-x/2 +3e^-x/2/2 which when integrating gives ye^-x/2 = -∫xe^-x/2dx + ∫3e^-x/2dx/2 +C. Use integration by parts to integrate ∫xe^-x/2dx. Let u=x and dv = e^-x/2dx. Now du=dx and v= -2 e^-x/2 so intregral is -2xe^-x/2 + 2∫e^-x/2dx = -2xe^-x/2 -4e^-x/2 so the integral is now 2xe^-x/2 +4e^-x/2. So ye^-x/2 =e^-x/2(2x +4) -3e^-x/2 +C. Now divide both sides by e^-x/2 to get y=2x+1 + Ce^x/2.

Use separation of variables for the second equation such that dy/y² = dx(1+e^x). Just integrate both sides. ∫y^-2dy = ∫(1+ e^x)dx. Doing the integration gives - y^-1= x +e^x + C. So 1/y = C -x-e^x. Therefore y= 1/(C-x-e^x).