# Thread: Help with using integrating factor to solve dy/dt = 2x + 4t (ans: -2t -1 + ce^(2t))

1. ## Help with using integrating factor to solve dy/dt = 2x + 4t (ans: -2t -1 + ce^(2t))

Hi,

First post here, so bear with me if I'm breaking forum etiquette

I'm working out exercises in my Croft & Davis engineering mathematics book. I'm supposed to solve the equation :

dy/dt = 2x + 4t

By using an integrating factor. According to the book, the answer should be :

-2t -1 + ce2t

While my answer is just -2t + ce2t. What am I doing wrong? And where does the -1 come from?

Thank you!

2. Originally Posted by MisterMD
I'm supposed to solve the equation :

dy/dt = 2x + 4t

By using an integrating factor. According to the book, the answer should be :

-2t -1 + ce2t

While my answer is just -2t + ce2t. What am I doing wrong? And where does the -1 come from?
Unfortunately, it is not possible to troubleshoot work that we cannot see. So please reply with a clear listing of all of your steps. Thank you!

3. Originally Posted by MisterMD
Hi,

First post here, so bear with me if I'm breaking forum etiquette

I'm working out exercises in my Croft & Davis engineering mathematics book. I'm supposed to solve the equation :

dy/dt = 2x + 4t
Is this the correct ODE? You are showing 3 variables!

4. Ah, sorry for not replying. I actually figured it out though! Again, I hope to make it clear here. Bear with me. Also thanks to Wolfram Alpha.

It's solving the equation by using an integrating factor. The first thing I did was then make it into the standard form dy/dx + P(x) = Q(x). So I turned dx/dt = 2x + 4t into dx/dt - 2x = 4t.

P is then -2, Q is 4t.

Then I worked out the integrating factor (called U) : e(integral(P) dx) which worked out to e-2t.

Multiplying everything with this factor makes it into this : (dx/dt * e-2t) - (2xe-2t) = (4te-2t). Using the product rule, that can be simplified to :

d(xe-2t)/dt = 4te-2t

Integrate both to produce : xe-2t = integral(4te-2t)dt, which is rather hairy. I did it with integration by parts.

4 (integral(te-2t)dt. Removing the factor of 4. Integration by parts is : Integral(UV') = UV - Integral(U'V)

U = t, U' = 1
V = (-e-2t)/2, V' = e-2t

So it becomes : (-te-2t)/2 - Integral((-e-2t)/2)

Becoming : (-te-2t)/2 - (e-2t)/4 + C

Rmoving fractions (multiplying all by 4) : -2te-2t - e-2t + C

And then the final step which is dividing all by U : x = -2t -1 + Ce2t which is exactly the solution in the book. Still! Thanks for replying