dydxxyyx y14 linear homogenous substitutionSolutionreplace u

dy/dx=(x/y+y/x), y(1)=-4 (linear homogenous substitution)

Solution

replace

u = x/y

u\' = (y - xy\')/y^2 = xu(1-uy\') = xu - y\'xu^2

y\' = (xu-u\')/(xu^2)

So your ODE becomes

(xu-u\')/(xu^2) = u + 1/u +1

xu - u\' = xu^3 + xu + xu^2

u\' = -xu^2(1+u)

u\'/{u^2(1+u)} = -x

Now you have your variables seperated

[1/(1+u) - (u-1)/u^2 ] du = -x dx

ln(1+u) - ln u - u^(-1) + ln C = - x^2

ln[C(1+u)/u] = -x^2 + 1/u

C(1+u)/u = e^(1/u)/e^(x^2)

u/(1+u) * e^(1/u) = C * e^(x^2)

x/(x+y) * e^(y/x) = C * e^(x^2)