obrozenecy6

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. $\left(2x-3y\right)dx+\left(2y-3x\right)dy=0$

### Answer & Explanation

Archie Jones

Classsify the following differential eq.
$\left(2x-3y\right)dx+\left(2y-3x\right)dy=0$
Solution:
$dy\left(2y-3x\right)=-\left(2x-3y\right)dx$
$\frac{dy}{dx}=-\frac{\left(2x-3y\right)}{\left(2y-3x\right)}$
Finding homogeneous
$F\left(\lambda x,\lambda y\right)=\frac{-\left(2\lambda x-3\lambda y\right)}{\left(2\lambda y-3\lambda x\right)}$
$=\frac{-\lambda \left(2x-3y\right)}{\left(2y-3x\right)}$
$=\frac{-\left(2x-3y\right)}{\left(2x-3x\right)}$
$={\lambda }^{\circ }F\left(r,y\right)$
The given equation is homogenous.

Robert Pina

Simplifying
$\left(2x+-3y\right)\cdot dx+\left(2y+-3x\right)\cdot dy=0$
Reorder the terms for easier multiplication:
$dx\left(2x+-3y\right)+\left(2y+-3x\right)\cdot dy=0$
$\left(2x\cdot dx+-3y\cdot dx\right)+\left(2y+-3x\right)\cdot dy=0$
Reorder the terms:
$\left(-3dxy+2{dx}^{2}\right)+\left(2y+-3x\right)\cdot dy=0$
$\left(-3dxy+2{dx}^{2}\right)+\left(2y+-3x\right)\cdot dy=0$
Reorder the terms:
$-3dxy+2{dx}^{2}+\left(-3x+2y\right)\cdot dy=0$
Reorder the terms for easier multiplication:
$-3dxy+2{dx}^{2}+dy\left(-3x+2y\right)=0$
$-3dxy+2{dx}^{2}+\left(-3x\cdot dy+2y\cdot dy\right)=0$
$-3dxy+2{dx}^{2}+\left(-3dxy+2{dy}^{2}\right)=0$
Reorder the terms:
$-3dxy+-3dxy+2{dx}^{2}+2{dy}^{2}=0$
Combine like terms: $-3dxy+-3dxy=-6dxy$
$-6dxy+2{dx}^{2}+2{dy}^{2}=0$
Solving

karton

$2x-3y+\left(2y-3x\right)\frac{dy}{dx}=0$
$2x-3y+\left(2y-3x\right){y}^{\prime }=0$
Verify that $\frac{\partial M\left(x,y\right)}{\partial y}=\frac{\partial N\left(x,y\right)}{\partial x}$: True
Find $\left(x,y\right):\left(x,y\right)={y}^{2}-3xy+{x}^{2}+{c}_{1}$
${y}^{2}-3xy+{x}^{2}+{c}_{1}={c}_{2}$
${y}^{2}-3xy+{x}^{2}={c}_{1}$
Isolate y: $y=\frac{3x+\sqrt{5{x}^{2}+4{c}_{1}}}{2},y=\frac{3x-\sqrt{5{x}^{2}+4{c}_{1}}}{2}$
$y=\frac{3x+\sqrt{5{x}^{2}+{c}_{1}}}{2},y=\frac{3x-\sqrt{5{x}^{2}+{c}_{1}}}{2}$

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