Solve the differential equation?

Question

Solve the differential equation y"+y'=e^t. 

 

Please show all your work step by step and tell me the right answer. Also tell me how to find the answer for this problem on Ti-89 calculator. Also tell me where can I find lots of problems like this problem.

Answer 1

Hi Sun,

 

This is a second order linear equation of the form

 

y''+p(t)y'=g(t)

 

with p(t) = 1 and g(t) = e^t

 

To solve the equation, we use the integrating factor method:

 

 

 

First we let u = y'. Then, u' = y''. Our equation is now a first order linear differential equation of the form

 

u' + p(t)u=g(t)

 

with the same p(t) and g(t) as before.

 

 

 

Next, we use our integrating factor μ(t). We set

 

μ(t) = e^{∫p(t) dt}

 

      = e^{∫1 dt}

    

      = e^t

 

Then, we solve for u(t):

 

u(t) = (∫μ(t)g(t) dt)/μ(t)

 

       = (∫e^t*e^t dt)/e^t

 

       = (∫e^2t dt)/e^t

 

       = ( (1/2)e^2t )/e^t

 

       = (1/2)e^t

 

Then, our solution is given by:

 

y(t) = ∫u(t) dt

 

       = ∫(1/2)e^t

 

       = (1/2)e^t

 

We can check this solution easily:

 

y''+y' = (1/2)e^t + (1/2)e^t

    

         = e^t

 

which satisfies our differential equation.

 

 

As for solving this on the TI-89, I am not sure I can help you there.

 

For similar problems, google "second order differential equations practice problems" and you should come up with plenty.