Answer to Question #294528 in Differential Equations for Raima

Question #294528

( x - 2ey ) dy + y + ( xsin(x) ) dx=0


1
Expert's answer
2022-02-07T16:33:18-0500
"P(x,y)=y+x\\sin x, \\dfrac{\\partial P}{\\partial y}=1"

"Q(x,y)=x-2e^y, \\dfrac{\\partial Q}{\\partial x}=1"

"\\dfrac{\\partial P}{\\partial y}=1=\\dfrac{\\partial Q}{\\partial x}"

Write the system of two differential equations that define the function "u(x,y)"


"\\dfrac{\\partial u}{\\partial x}=y+x\\sin x"

"\\dfrac{\\partial u}{\\partial y}=x-2e^y"

"u=\\int(x-2e^y)dy+\\varphi(x)"

"u=xy-2e^y+\\varphi(x)"


"\\dfrac{\\partial u}{\\partial x}=y+\\varphi'(x)=y+x\\sin x"

"\\varphi'(x)=x\\sin x"


"\\varphi(x)=\\int x\\sin xdx"

"\\varphi(x)=-x\\cos x+\\sin x+C"


"u=xy-2e^y-x\\cos x+\\sin x+C"

"-xy+2e^y+x\\cos x-\\sin x=C"



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