solve: x dy/dx=2xe^x-y+6x^2 → (y-6x^2-2xe^x) dx+ xdy=0

x dy/ dx=2xe^x-y+6x^2 → (y-6x^2-2xe^x) dx+xdy=0
in Calculus Answers by Level 12 User (101k points)

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15 Answers

P_= y-6x^2-2xe^x
by Level 12 User (101k points)
Q_= x and æp/æy=1
by Level 12 User (101k points)
=æQ/æx therefore the given is exact :
by Level 12 User (101k points)
du_= Pdx+ Qdy=0
by Level 12 User (101k points)
u(x,y)=c
by Level 12 User (101k points)
æu/æy=x
by Level 12 User (101k points)
U=xy+w(x)
by Level 12 User (101k points)
And æu/æx=y+w'(x)_=y-6x^2-2xe^x
by Level 12 User (101k points)
Then w(x)
by Level 12 User (101k points)
=-{(6x^2+2xe^x) dx
by Level 12 User (101k points)
=-2x^3-2{xd(e^x)
by Level 12 User (101k points)
=-2x^3-2(xe^x-e^x)
by Level 12 User (101k points)
W(x)=-2x^3-2xe^x+2e^x
by Level 12 User (101k points)
Therefore xy=2c^3+2xe^x-2e^x+c
by Level 12 User (101k points)
Y=2x^2+2•(1-1/x)e^x+c/x
by Level 12 User (101k points)

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