hvacwk

Answered

2021-12-29

Determine whether the following differential equations are exact or not exact. if exact, solve for the general solution

$(x+2y)dx-(2x+y)dy=0$

Answer & Explanation

turtletalk75

Expert

2021-12-30Added 29 answers

Step 1

$(x+2y)dx-(2x+y)dy=0$

The differential equation$Mdx+Ndy=0$ is a exact if

$\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$

Step 2

From the given differential equation

$M=x+2y$

$N=-(2x+y)$

$N=-(2x+y)$

$\frac{\partial M}{\partial y}=2$

$\frac{\partial N}{\partial x}=-2$

Since,$\frac{\partial M}{\partial y}\ne \frac{\partial N}{\partial x}$

The given differential equation is not exact differential equation.

The differential equation

Step 2

From the given differential equation

Since,

The given differential equation is not exact differential equation.

Buck Henry

Expert

2021-12-31Added 33 answers

Simplifying

$(x+2y)\cdot dx+-1(2x+-1y)\cdot dy=0$

Reorder the terms for easier multiplication:

$dx(x+2y)+-1(2x+-1y)\cdot dy=0$

$(x\cdot dx+2y\cdot dx)+-1(2x+-1y)\cdot dy=0$

Reorder the terms:

$(2dxy+{dx}^{2})+-1(2x+-1y)\cdot dy=0$

$(2dxy+{dx}^{2})+-1(2x+-1y)\cdot dy=0$

Reorder the terms for easier multiplication:

$2dxy+{dx}^{2}+-1dy(2x+-1y)=0$

$2dxy+{dx}^{2}+(2x\cdot -1dy+-1y\cdot -1dy)=0$

$2dxy+{dx}^{2}+(-2dxy+1{dy}^{2})=0$

Reorder the terms:

$2dxy+-2dxy+{dx}^{2}+1{dy}^{2}=0$

Combine like terms:$2dxy+-2dxy=0$

$0+{dx}^{2}+1{dy}^{2}=0$

${dx}^{2}+1{dy}^{2}=0$

Solving

${dx}^{2}+1{dy}^{2}=0$

Solving for variable d.

Move all terms containing d to the left, all other terms to the right.

Factor out the Greatest Common Factor (GCF), d.

$d({x}^{2}+{y}^{2})=0$

Reorder the terms for easier multiplication:

Reorder the terms:

Reorder the terms for easier multiplication:

Reorder the terms:

Combine like terms:

Solving

Solving for variable d.

Move all terms containing d to the left, all other terms to the right.

Factor out the Greatest Common Factor (GCF), d.

karton

Expert

2022-01-10Added 439 answers

It is a homogeneous equation

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