guringpw

2021-12-31

Solve the following manually using separable differential equations method.
$mydx+nxdy=0$

### Answer & Explanation

MoxboasteBots5h

$mydx+nxdy=0$
$mydx=-nxdy$
$\frac{dx}{x}=\frac{-n}{m}\cdot \frac{dy}{y}$
Take both side integration:
$\int \frac{dx}{x}=\frac{-n}{m}\int \frac{dy}{y}$
$\mathrm{ln}\left(x\right)=\frac{-n}{m}\mathrm{ln}\left(y\right)+c$
$\left(\therefore c=const\right)$
Answer: $\mathrm{ln}\left(x\right)+\frac{n}{m}\mathrm{ln}\left(y\right)=c$

Thomas Nickerson

Simplifying
$mydx+-1nxdy=0$
Solving
$dmxy+-1dnxy=0$
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), 'dxy'.
$dxy\left(m+-1n\right)=0$
Subproblem 1
Set the factor 'dxy' equal to zero and attempt to solve:
Simplifying
$dxy=0$
Solving
$dxy=0$
Move all terms containing d to the left, all other terms to the right.
Simplifying
$dxy=0$
The solution to this equation could not be determined.
This subproblem is being ignored because a solution could not be determined.
Subproblem 2
Set the factor ${}^{\prime }{\left(m+-1n\right)}^{\prime }$ equal to zero and attempt to solve:
Simplifying
$m+-1n=0$
Solving
$m+-1n=0$
Move all terms containing d to the left, all other terms to the right.
Add '-1m' to each side of the equation.
$m+-1m+-1n=0+-1m$
Combine like terms: $m+-1m=0$
$0+-1n=0+-1m$
$-1n=0+-1m$
Remove the zero:
$-1n=-1m$
Add 'n' to each side of the equation.
$-1n+n=-1m+n$
Combine like terms: $-1n+n=0$
$0=-1m+n$
Simplifying
$0=-1m+n$
The solution to this equation could not be determined.
This subproblem is being ignored because a solution could not be determined.
The solution to this equation could not be determined.

Vasquez

$mydx=nxdy$
$⇒my=nx\frac{dy}{dx}$
Substitute $\frac{dy}{dx}$ with $y\text{'}$.
$⇒my=nxy\text{'}$
$⇒\frac{1}{y}y\text{'}=\frac{m}{nx}$
$⇒\mathrm{ln}y=\frac{m\mathrm{ln}\left(x\right)}{n}+{c}_{1}$
Answer: $⇒y=e\frac{m\mathrm{ln}\left(x\right)}{n}+{c}_{1}$

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