Identify the form of the nonlinear second order differential equation

Question

Identify the form of the nonlinear second order differential equation: $y''y'-x^2=0$, on the domain $ \text D[0,\infty)$ and thus find soln. given $y(0)=y'(0)=0$

Just a question about how one would solve $y(x)$ given the equation above. I used subsitution in that some $v(x) = y'(x)$ and solved that way but I do not know how to apply the intial conditions to what I get for $y(x)$ which is in an unsolvable interal form.

So I believe that there is another way of approach, I tried the taylor series method but I believe I am lost in that.

Just a quick explaination is all, I am not too confident on the taylor series expansion if that is the correct way.

Thanks

Answer 1

Notice how $$\frac d{dx}\frac{y’^2}2=\frac12 2y’\frac d{dx} y’=y’y’’$$

using the chain rule. Then, integrate both sides to get:$$\frac12 y’^2(x)-\frac13 x^3+c_1=0$$

Now solve for $y’$ and integrate by doing $$y=\pm \int \sqrt{\frac13x^3+c_1}dx=\pm\frac1{\sqrt 3}\int \sqrt{c_1+x^3}dx$$

which is a Gauss Hypergeometric function.

Here is proof of the hypergeometric function result:

Arc length of $x^n$ found using Hypergeometric function and series.

Which can be simplified via Incomplete Beta function. I will leave the rest up to you. Please correct me and give me feedback!