Classify the following differential equations as separable, homogeneous, parallel line, or exact. Explain briefly your...

obrozenecy6

obrozenecy6

Answered

2021-12-31

Classify the following differential equations as separable, homogeneous, parallel line, or exact. Explain briefly your answers. Then, solve each equation according to their classification. (2x3y)dx+(2y3x)dy=0

Answer & Explanation

Archie Jones

Archie Jones

Expert

2022-01-01Added 34 answers

Classsify the following differential eq.
(2x3y)dx+(2y3x)dy=0
Solution:
dy(2y3x)=(2x3y)dx
dydx=(2x3y)(2y3x)
Finding homogeneous
F(λx,λy)=(2λx3λy)(2λy3λx)
=λ(2x3y)(2y3x)
=(2x3y)(2x3x)
=λF(r,y)
The given equation is homogenous.
Robert Pina

Robert Pina

Expert

2022-01-02Added 42 answers

Simplifying
(2x+3y)dx+(2y+3x)dy=0
Reorder the terms for easier multiplication:
dx(2x+3y)+(2y+3x)dy=0
(2xdx+3ydx)+(2y+3x)dy=0
Reorder the terms:
(3dxy+2dx2)+(2y+3x)dy=0
(3dxy+2dx2)+(2y+3x)dy=0
Reorder the terms:
3dxy+2dx2+(3x+2y)dy=0
Reorder the terms for easier multiplication:
3dxy+2dx2+dy(3x+2y)=0
3dxy+2dx2+(3xdy+2ydy)=0
3dxy+2dx2+(3dxy+2dy2)=0
Reorder the terms:
3dxy+3dxy+2dx2+2dy2=0
Combine like terms: 3dxy+3dxy=6dxy
6dxy+2dx2+2dy2=0
Solving
karton

karton

Expert

2022-01-09Added 439 answers

2x3y+(2y3x)dydx=0
2x3y+(2y3x)y=0
Verify that M(x,y)y=N(x,y)x: True
Find (x,y):(x,y)=y23xy+x2+c1
y23xy+x2+c1=c2
y23xy+x2=c1
Isolate y: y=3x+5x2+4c12,y=3x5x2+4c12
y=3x+5x2+c12,y=3x5x2+c12

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