Solve differential equationu'-5u=ve^(-5v)

I have been having a problem with this simple equation. It is asking me this: Find all values of $k$ for which the function $y=\sin (kt)$ satisfies the differential equation $y\prime \prime +7y=0$.
I have found the second derivative, plugged it back into the differential equation, and found out $\sqrt{7}$ checked, but $-\sqrt{7}$ did not. Please help me solve this question.

Find the solution ${\displaystyle \left(x\frac{\sin y}{x}\right)dy=(y\frac{\sin y}{x}-x)dx}$

${\displaystyle 2\cdot \left(\frac{dy}{dx}\right)+2y=0}$

Consider the following second-order linear homogeneous difference equation with constant coefficients and two initial conditions:
${\displaystyle u_{n+2}-6u_{n+1}+9u_n=0,u_0=1,u_1=9}$
Determine the sequence ${\displaystyle u_2,u_3,u_4,u_5}$. Solve the difference equation for ${\displaystyle u_n}$ and use the result to check for ${\displaystyle u_5}$

Write first-order differential equations that express the situation: A Walking Mode, l Construct a mathematical model to estimate how long it will take you to walk to the store for groceries. Suppose you have measured the distance as 11 mile and you estimate your walking speed as 33 miles per hour. If it takes you 2020 minutes to get to the store, what do you conclude about your model?

Using the Laplace Transform Table in the textbook and Laplace Transform Properties,find the (unilateral) Laplace Transforms of the following functions:
${\displaystyle t\cos \left({\omega }_{0}t\right)u\left(t\right)}$

Find the Laplace transform $L\{u_3(t)(t^2-5t+6)\}$
$a)F(s)=e^{-3s}(\frac{2}{s^4}-\frac{5}{s^3}+\frac{6}{s^2})$
$b)F(s)=e^{-3s}(\frac{2}{s^3}-\frac{5}{s^2}+\frac{6}{s})$
$c)F(s)=e^{-3s}\frac{2+s}{s^4}$
$d)F(s)=e^{-3s}\frac{2+s}{s^3}$
$e)F(s)=e^{-3s}\frac{2-11s+30s^2}{s^3}$