Classify the following differential equations as separable, homogeneous, parallel line, or exact. Explain briefly your answers. Then, solve each equation according to their classification. (2x−3y)dx+(2y−3x)dy=0

Name: Classify the following differential equations as separable, homogeneous, parallel line, or exact. Explain briefly your answers. Then, solve each equation according to their classification. (2x−3y)dx+(2y−3x)dy=0
Uploaded: 2022-02-23T17:22:57+00:00
Description: Classify the following differential equations as separable, homogeneous, parallel line, or exact. Explain briefly your answers. Then, solve each equation according to their classification. (2x−3y)dx+(2y−3x)dy=0

Question

Archie Jones · Accepted Answer

Classsify the following differential eq.  (2x−3y)dx+(2y−3x)dy=0  Solution:  dy(2y−3x)=−(2x−3y)dx  dydx=−(2x−3y)(2y−3x)  Finding homogeneous  F(λx,λy)=−(2λx−3λy)(2λy−3λx)  =−λ(2x−3y)(2y−3x)  =−(2x−3y)(2x−3x)  =λ∘F(r,y)  The given equation is homogenous.

Robert Pina · Accepted Answer

Simplifying   (2x+−3y)⋅dx+(2y+−3x)⋅dy=0   Reorder the terms for easier multiplication:   dx(2x+−3y)+(2y+−3x)⋅dy=0   (2x⋅dx+−3y⋅dx)+(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+(−3x⋅dy+2y⋅dy)=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 · Accepted Answer

2x−3y+(2y−3x)dydx=02x−3y+(2y−3x)y′=0Verify that ∂M(x,y)∂y=∂N(x,y)∂x: TrueFind (x,y):(x,y)=y2−3xy+x2+c1y2−3xy+x2+c1=c2y2−3xy+x2=c1Isolate y: y=3x+5x2+4c12,y=3x−5x2+4c12y=3x+5x2+c12,y=3x−5x2+c12