Question

solve x dy/dx + y= y²​

Answer 1

Step-by-step explanation:

Let y = xv … (1) [v is a function of x] —> dy/dx = x (dv/dx) + v … (2)

Substitute (1) and (2) in the given differential equation dy/dx=(y-x) /(x+y):

x (dv/dx) + v = (xv - x)/(xv + x) —> x (dv/dx) = {(v - 1)/(v + 1)} - v —>

x (dv/dx) = - (v^2 + 1)/(v + 1) —> {(v/(v^2 + 1) + 1/(v^2 + 1)} dv = - dx/x

Integrate:(1/2) ln Iv^2 + 1I + arctan(v) = -ln IxI + C —>

ln {sqrtIv^2 + 1I} + ln IxI + arctan(v) = C —>

ln IxI sqrt{(y^2/x^2) + 1} + arctan(y/x) = C —>

ln [sqrt(x^2 + y^2)] + arctan Iy/xI = C