Help me with Calculus question?

Question

At any time t≥0, the velocity of a particle traveling along the x-axis is given by the differential equation dx/dt-10x=60e^4t. 

 

Find the general solution x(t) for the position of the particle. 

 

Please show all your work. 

Answer 1

dx/dt – 10x = 60e^(4t)

According to Paul, this is a linear first order differential equation (see http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx).

We need to multiply both sides by the “integrating factor”:

u(t) = e^(∫(-10 dt)) = e^(-10t)

e^(-10t)x’ – 10x e^(-10t) = 60e^(4t) e^(-10t)

 

The left side is the result of a product rule so reverse it:

(x e^(-10t))’ = 60e^(-6t)

Integrate both sides:

x e^(-10t) = 60e^(-6t)/(-6) + C

x = –10e^(4t) + Ce^(10t)

 

Because of C this is a family of solutions.

check:

x’ = –40e^(4t) + 10Ce^(10t)

If x = –10e^(4t) + Ce^(10t), then
Ce^(10t) = x + 10e^(4t)

x’ = –40e^(4t) + 10(x + 10e^(4t))

x’ = –40e^(4t) + 10x + 100e^(4t))

x’ = 60e^(4t) + 10x

x’ – 10x = 60e^(4t) √