Find the solutions to:$\frac{d^2y}{dx^2}=\left(\frac{dy}{dx}\right)^2$.

Question

Find the solutions to:$\displaystyle\frac{d^2y}{dx^2}=\left(\frac{dy}{dx}\right)^2$.

I got the following solutions:-

$\left(\frac{dy}{dx}\right)=0\Rightarrow y=c_1$ is a solution

$\left(\frac{dy}{dx}\right)=1\Rightarrow y=x+c_2$ is another solution

Are there any other solutions?

I dont have any idea about how to solve a $2^{nd}$ order non linear DE. As far as i Know , a $2^{nd}$ order linear DE could be solved with the help of auxillary equations , Is there any such similar methods applicable to this problem

Answer 1

Notice there is no 0th order derivative here. Hence, this is actually just a first-order equation in disguise. Substitute $v = \frac{dy}{dx}$ $$ \frac{dv}{dx} = v^2 $$

Separate this and solve

$$ v(x) = \frac{1}{c_1-x} $$

Then integrate back

$$ y(x) = c_2 -\ln|c_1-x| $$

Answer 2

Hint: Let $p(x) = \frac{dy}{dx}$. Then: $$ p'(x) = p(x)^2 $$ is a first order DE.

Answer 3

Substitute $$\frac{dy(x)}{dx}=v(x)$$ and then you will get $$\frac{\frac{dv(x)}{dx}}{v(x)^2}=1$$