I would like to solve:
In order to do so, we let , and we assume x as a function of t. Now, we take derivative with respect to t from the differential equation, and obtain
By the chain rule, we have: . So, the above simplifies to
That is, we have: . Thus, we obtain
Now, if we want to verify the solution, it turns out that C must be zero, in other words, satisfies the original differential equation.
I have two questions:
1) What happens to the integration constant? That is, what is the general solution of the differential equation?
2) If we try to solve this differential equation with Mathematica, we obtain
which has a different form from the analytical approach. How can we also produce this result analytically?