Solve the exact differential equation (2xy-x^-2*y^-2)dx+(x^2-2x^-1*y^-3)dy=0

Solve the exact differential equation (2xy-x^-2*y^-2)dx+(x^2-2x^-1*y^-3)dy=0
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20 Answers

The given is of the form:
by Level 12 User (101k points)
Pdx+Qdy=0
by Level 12 User (101k points)
æp/æy=2x-x^-2(-2y^-3)
by Level 12 User (101k points)
=2x+2x^-2•y^-3
by Level 12 User (101k points)
Also æQ/æx=æp/æy
by Level 12 User (101k points)
That is, the given is exact:
by Level 12 User (101k points)
du_=Pdx+Qdy=0
by Level 12 User (101k points)
U(x,y)=c1
by Level 12 User (101k points)
æu/æy= Q
by Level 12 User (101k points)
→ u=x^2y-2/x•(-1/2y^2)+w(x)
by Level 12 User (101k points)
→ u=w(x)+x^2y+1/xy^2
by Level 12 User (101k points)
æu/æx=w'(x)+2xy-(1/xy)^2_=P
by Level 12 User (101k points)
→ w'(x)=0
by Level 12 User (101k points)
And w(x)=c2
by Level 12 User (101k points)
Therefore:x^2y+x^-1 y^-2+c2=c1
by Level 12 User (101k points)
→ x^2 y+1/xy^2=c1-c2
by Level 12 User (101k points)
→ x^3 y^3+1+(c2-c1)xy^2=0
by Level 12 User (101k points)
And division by x^3 gives
by Level 12 User (101k points)
Y^3+(c/x^2) y^2+(1/x^3)=0
by Level 12 User (101k points)
( incomplete cubic in y)
by Level 12 User (101k points)

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