Solve the following equations x2dydx yxy dydx y21 dydx y

Solve the following equations x2dy/dx = y-xy dy/dx= y2=1 dy/dx + y = x, y(0) = 4. (x+y)2 dx + )2xy +x2-1) day = 0, y(1) = 1.

Solution

a) Rearrange in the form dy/y = (1-x)dx/x^2 integrating we get ln y = -1/x - ln x + c ln xy = -1/x + c xy = ke^(-1/x) b)dy/(y^2-1) = dx integrate (1/2)ln((y-1)/(y+1)) = x + c c) Use integrating factor as e^x and then solve this equation y.e^x = integration of (e^x*x) dx + c d)