Given an ordinary differential equation (O.D.E.):
$$y\left(\frac{d^5y}{dx^5}\right)^2+x^3\left(\frac{d^3y}{dx^3}\right)^3+x\left(\frac{dy}{dx}\right)=y^2\sin(x^2)$$
Which of the following options is correct for given O.D.E.?
a) It is a fifth order linear O.D.E. of degree 2
b) It is a fifth order non-linear O.D.E. of degree 3
c) It is a fifth order non-linear O.D.E. of degree 2
d) None
My try:
Given D.E. $y\left(\frac{d^5y}{dx^5}\right)^2+x^3\left(\frac{d^3y}{dx^3}\right)^3+x\left(\frac{dy}{dx}\right)=y^2\sin(x^2)$
The order of O.D.E.=highest order=5
The degree of O.D.E.=degree of highest order derivative =2
Since $y$ is multiplied with $\left(\frac{d^5y}{dx^5}\right)$ or $y$ has a power $2$ so it is a non-linear O.D.E.
Thus, My answer becomes (c) but I am not sure if i am right. Please tell me if I am wrong or right & correct me if i am wrong & give explanation.
Thank you.
$$y\left(\frac{d^5y}{dx^5}\right)^2+x^3\left(\frac{d^3y}{dx^3}\right)^3+x\left(\frac{dy}{dx}\right)=y^2\sin(x^2)$$
is a fifth order non-linear O.D.E. of degree $2$ so your answer is correct.
You have explained each step very clearly.