How to solve for this differential Equations Analytically

[Marwanx]

Hello,

Could you please help me to solve the differential equations analytically?

dvy/dt = - (k/m) * vy - g

dy/dt = vy

vy = vo*sin(theta) = vy, y = yo

[HallsofIvy]

Hello,

Could you please help me to solve the differential equations analytically?

dvy/dt = - (k/m) * vy - g

dy/dt = vy

vy = vo*sin(theta) = vy, y = yo
Are these last equations "intial values"- that is, values at t= 0?
If so, you should have written "vy(0)= vo*sin(theta)" and "y(0)= yo".

Since these are linear equations with constant coefficients, it should be easy to solve them. In fact, these are only "partially coupled". That is, the first equation involves only vy so you can immediately solve it. Write it as
\frac{dvy}{dt}= -(k/m)y -g
a linear equation of order 1 with constant coefficients. You can solve the "homogeneous"part first:
\frac{dvy}{dt}= -(k/m)vy
That's easy- that's a "separable" equation. After you have found the general solultion to that, just add a solution to the entire equation. I recommend trying something of the form vy= A, a constant. Put that into the equation and solve for A. Then add A to the solution to the homogeneous equation.

After you have found vy as a function of t, just put it into dy/dt= vy and integrate dy= vy dt to find y.