Solve the following IVP for the exact equation and find the interval of validity for...

widdonod1t

widdonod1t

Answered

2021-12-29

Solve the following IVP for the exact equation and find the interval of validity for the solution.
2xy9x2+(2y+x2+1) dydx=0 y(0)=2

Answer & Explanation

Barbara Meeker

Barbara Meeker

Expert

2021-12-30Added 38 answers

For any differential equation that is given to be exact, we must have a function f(x,y) such that:
d[f(x,y)]=fxdx+fydy=M dx+N dy
M(x,y)=fx and N(x,y)=fy
i.e.f(x,y)=M(x,y)dx+ϕ1(y) and f(x,y)=N(x,y)dy+ϕ2(x)
The given differential equation can be written as:
2xy9x2+(2y+x2+1)dydx=0
(2xy9x2)dx+(2y+x2+1)dy=0 (i)
Comparing to the standard form of M(x,y)dx+N(x.y)dy=0 we get:
M(x,y)=2xy9x2
f(x,y)=(2xy9x2)dx+ϕ1(y)
=2y(x22)9(x33)+ϕ1(y)
=x2y3x3+ϕ1(y)...(2)
N(x,y)=2y+x2+1
f(x,y)=(2y+x2+1)dy+ϕ2(x)
=2(y22)+x2(y)+(y)+ϕ2(x)
reinosodairyshm

reinosodairyshm

Expert

2021-12-31Added 36 answers

[y(2x)dx+x2(dy)]+(2y+1)dy9x2dx=0
d(x2y)+d(y2+y)d(3x3)=0
d(x2y+y2+y3x2)=0
x2y+y2+y3x2=c
y(0)=30+43+0=c=6x2y+y2+y3x2=6
karton

karton

Expert

2022-01-09Added 439 answers

(2xy9x2)dx+(2y+x2+1)dy=0My=2x=Nx=2xM dx+g(y)H(x,y)=2xy9x2dx+g(y)=2y+x2+1 dy=x2y3x3+g(y)=y2+x2y+y
Since x2y appears on both sides, cancel it on the yside, which gives g(y)=y2+y. Add this to the leftside and the general solution is:
H(x,y)=x2y3x2+y2+y

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