Find the solution of the following differential equations: Homogeneous Differential Equations ЗхуууЗху2dx+(3x2у+у3)dy=0;y(1)=2 (x2−9y2)dx+2xydy=0;y(2)=−1

hunterofdeath63

hunterofdeath63

Answered

2021-12-30

Find the solution of the following differential equations:
Homogeneous Differential Equations
Зху2dx+(3x2у+у3)dy=0;y(1)=2
(x29y2)dx+2xydy=0;y(2)=1

Answer & Explanation

Matthew Rodriguez

Matthew Rodriguez

Expert

2021-12-31Added 32 answers

The given homogenous differential equation
3xy2dx+(3x2y+y3)dy=0 (1)
y(1)=2 (2)
First we check the Exactness of the given Differential equation,
Mdx+Ndy=0
M=3xy2,N=3x2y+y3
Then, My=6xy,Nx=6xy
My=Nx
So, the given differential equation is exact.
Hence the solution is given by.
Mdx+(terms in N not containg x)dy=c
3xy2dx+y3dy=c
3y2x22+y44=c (3)
32x2y2+y44=c
using (2)
Put x=1,y=2
3(1)2(2)2+(2)44=c
6+4=cc=10
(3) becomes
32x2y2+y44=10
3x2y2+y4=40 This is the required solution.
einfachmoipf

einfachmoipf

Expert

2022-01-01Added 32 answers

(x29y2)dx+2xydy=0
y(2)=1
(x29y2)dx=2xydy
dydx=(x29y2)2xy
y=vx
dydx=v+xdvdx
v+xdvdx=(x29v2x2)2×x×vx
v+xdvdx=x2(19v2)2x2×v=(19v2)2v
xdvdx=1+9v22v22v=1+7v22v
2v7v21dv=dxx
Let 7v21=f
14bdv=dt2vdv=dt7
17dtt=dtx
17ln(t)=ln(x)+lnc
ln(7v21)17=ln(xc)
Vasquez

Vasquez

Expert

2022-01-09Added 457 answers

(1)3xy2dx+(3x2y+y3)dy=0(1)y(1)=2(2)First we check the Exactness of the given Differentialequation,Mdx+Ndy=0M=3xy2,N=3x2y+y3Then, My=6xy,Nx=6xyMy=

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?