Haiphongum

2022-09-15

Show substitution leads to a separable differential equation.
Consider the differential equation...
${y}^{\prime }=f\left(\frac{y}{t}\right)$
Show that the substitution $v=\frac{v}{t}$ leads to a separable differential equation in v
Here's what I did.
$v=\frac{y}{t}$
$\frac{dv}{dt}=\frac{dy}{dt}-\frac{1}{{t}^{2}}$
Sub into the orignal.
$\frac{dv}{dt}=f\left(v\right)-\frac{1}{{t}^{2}}$
$\frac{dv}{f\left(v\right)}=\frac{1}{{t}^{2}}dt$
This is where I get stuck. Is this the what the question is asking for, or am I forgetting to do something?

Claire Larson

Note that if you substitute $v=\frac{y}{t}⇒y=vt$. Note that v is assumed to be a function of t. Using the product rule, we obtain the derivative y′:
$\begin{array}{}\text{(1)}& y=v\left(t\right)\cdot t⇒{y}^{\prime }=\frac{dv}{dt}\cdot t+v\end{array}$
Substituting into ${y}^{\prime }=f\left(\frac{y}{t}\right)$:
$\frac{dv}{dt}\cdot t+v=f\left(v\right)$
Now, try to separate the variables. i.e, put only v and dv terms on one side of the equality and t and dt terms on the other side, then you are done.
Feel free to ask on the comments if you have any related doubts or questions.

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