Introduction to differential equations

Show that $x = e^{-2t}$ is a solution to $x'' + 4x' + 4x = 0$.

Is $y = x^2$ a solution to $x^2y'' - 2y = 0$? Justify.

Let $xy'' - y' = 0$. Try a solution of the form $y = x^r$. Is this a solution for some $r$? If so, find all such $r$.

Verify that $x=C_1e^t+C_2$ is a solution to $x''-x' = 0$. Find $C_1$ and $C_2$ so that $x$ satisfies $x(0) = 10$ and $x'(0) = 100$.

Solve $\frac{d\varphi}{ds} = 8 \varphi$ and $\varphi(0) = -9$.

Solve:

1. $\dfrac{dx}{dt} = -4x, \quad x(0)=9$
2. $\dfrac{d^2x}{dt^2} = -4x, \quad x(0)=1, \quad x'(0)=2$
3. $\dfrac{dp}{dq} = 3 p, \quad p(0)=4$
4. $\dfrac{d^2T}{dx^2} = 4 T, \quad T(0)=0, \quad T'(0)=6$