Question: x^2·y^3+x(1+y^2)y’=0 μ(x,y)=1/xy^3
Answer:
let x^2·y^3 be M(x,y) and x(1+y^2) be N(x,y)
M(y)=d(x^2y^3)/dy=3x^2·y^2
N(x)=d(x(1+y^2))/dx=1+y^2
M(y) is not equal to N(x), which means given equation is not an exact equation.
Next step, multiply both side of M(y) and N(x) with the integrating factor μ(x,y)=1/xy^3
1/xy^3(x^2·y^3+x(1+y^2)y’)=0
x+((1+y^2)/y^3)y’=0
let x be the M1(x,y) and (1+y^2)/y^3 be N1(x,y)
M1(x,y)=d(x)/dy=0
N1(x,y)=d((1+y^2)/y^3)/dx=0
As M1(x,y)=N1(x,y). Therefore, the equation is an exact equation.
∃φ(x,y) s.t. φx=M1
φ=∫M1dx=∫xdx=1/2 x^2+h(y)
φy=h(y)'=N1=(1+y^2)/y^3
h(y)=-1/2y^-2+ln|y|
φ=1/2 x^2-1/2y^-2+ln|y|=C
This is the general solution of this question
