I am unable to understand how to find the differential equation when a general solution has been given. Here are a few example solutions, which require their differential equations to be found:
(a) $y = ax^2 + bx + c$
(b) $y^2 = 4ax$
(c) $x^2 – 2xy + y^2 = a^2$
Since I have my test coming up, I would be grateful if someone could explain the logic of solving such a question. You could perhaps help me with 2 of the questions, and I will try the third one.
Hoping to receive some help soon.
Thank you
Best Answer
Remember that an expression with $n$ arbitrary constants will yield a differential equation of order $n$. So to get the $n^{th}$ order derivative you'll have to differentiate the expression $n$ times, and in that process you'll obtain $n$ more relations so that now you have a total of $n+1$ relations from which you can eliminate the $n$ arbitrary constants to obtain the differential equation.
Most of the times though the constants more or less dissappear by themselves. For example,consider $y=ax^2+bx+c$. There are 3 arbitrary constants $a$,$b$ and $c$ so just differentiate 3 times to obtain the DE $y'''=0$
Now consider $y^2=4ax$. Since there is only one constant $a$, differentiate once to get $2yy'=4a$. Now eliminate $4a$ to obtain the DE $2xy'=y$
I think with that in mind you can find the DE for a given solution.