D.E. Linear equation with integrating factor

[Marco Lugo]

Homework Statement

https://webwork.utpa.edu/webwork2_files/tmp/equations/2d/02a7e6a06f5b2424758fa01cc965f71.png [Broken] with https://webwork.utpa.edu/webwork2_files/tmp/equations/80/81c176aa8964438a63eb096513245f1.png [Broken]

Homework Equations

[/B]
Standard form: y' + p(x)y = f(x)

Integrating factor: r(x)= e^∫p(x)dx

Solution: y(x) = e^-∫p(x)dx(∫e^∫p(x)dx ⋅ f(x) dx + C)

The Attempt at a Solution

From, y' - y = 7e^t + 56e^8t

I got, p(x) = -1 and f(x) = 7e^t + 56e^8t

So the integrating factor is r(t)= e^{∫-1 dt} = e^-t

Plug it into the solution formula,
y(t)= e^t [∫e^-t (7e^t + 56e^8t) dt + C]

= e^t (7t + 8e^7t+ C)

= 7te^t + 8e^8t+ Ce^t

Using the initial condition y(0)= 2

= 8 + C = 2
C = -6

My final solution,
y(t) = 7te^t + 8e^8t- 6e^t

check: y(0) = 7(0)e⁽⁰⁾ + 8e⁸⁽⁰⁾- 6e⁽⁰⁾ = 2

But when I type it into WeBWork, it says my answer is incorrect. I don't think its a syntax problem because it has a preview button that allows you to see your actual problem.

Could it be that, y' - y = 7e^t + 56e^8t is not in standard form?

 

[Mark44]

Homework Statement

https://webwork.utpa.edu/webwork2_files/tmp/equations/2d/02a7e6a06f5b2424758fa01cc965f71.png [Broken] with https://webwork.utpa.edu/webwork2_files/tmp/equations/80/81c176aa8964438a63eb096513245f1.png [Broken]

Homework Equations

[/B]
Standard form: y' + p(x)y = f(x)

Integrating factor: r(x)= e^∫p(x)dx

Solution: y(x) = e^-∫p(x)dx(∫e^∫p(x)dx ⋅ f(x) dx + C)

The Attempt at a Solution

From, y' - y = 7e^t + 56e^8t

I got, p(x) = -1 and f(x) = 7e^t + 56e^8t

So the integrating factor is r(t)= e^{∫-1 dt} = e^-t

Plug it into the solution formula,
y(t)= e^t [∫e^-t (7e^t + 56e^8t) dt + C]

= e^t (7t + 8e^7t+ C)

= 7te^t + 8e^8t+ Ce^t

Using the initial condition y(0)= 2

= 8 + C = 2
C = -6

My final solution,
y(t) = 7te^t + 8e^8t- 6e^t

check: y(0) = 7(0)e⁽⁰⁾ + 8e⁸⁽⁰⁾- 6e⁽⁰⁾ = 2

But when I type it into WeBWork, it says my answer is incorrect. I don't think its a syntax problem because it has a preview button that allows you to see your actual problem.

Could it be that, y' - y = 7e^t + 56e^8t is not in standard form?

I don't see anything wrong with your work or your solution. You can (and should) verify for yourself that your solution is correct by first checking that the initial condition is satisfied, and then by substituting your solution into the differential equation. Your solution should make the DE a true statement.

As far as why the software doesn't accept your solution, make sure that you are working the same problem it thinks you are working. I would also check with the instructor.

 

[Marco Lugo]

I don't see anything wrong with your work or your solution. You can (and should) verify for yourself that your solution is correct by first checking that the initial condition is satisfied, and then by substituting your solution into the differential equation. Your solution should make the DE a true statement.

As far as why the software doesn't accept your solution, make sure that you are working the same problem it thinks you are working. I would also check with the instructor.

I checked the initial condition but you're right I should have substituted my solution into the problem.

my solution,
y(t) = 7te^t + 8e^8t - 6e^t

check:
dy/dt= 64e^8t + 7te^t + e^t

substituting both into the initial probelm,

dy/dt - y = 56e^8t + 7e^t

64e^8t + 7te^t + e^t - 7te^t - 8e^8t + 6e^t

I get, 56e^8t + 7e^t

I don't see anything wrong. I would submit the answer again but I only have 2 more tries, so I want to get some more advice before I do.

 

[Ray Vickson]

I checked the initial condition but you're right I should have substituted my solution into the problem.

my solution,
y(t) = 7te^t + 8e^8t - 6e^t

check:
dy/dt= 64e^8t + 7te^t + e^t

substituting both into the initial probelm,

dy/dt - y = 56e^8t + 7e^t

64e^8t + 7te^t + e^t - 7te^t - 8e^8t + 6e^t

I get, 56e^8t + 7e^t

I don't see anything wrong. I would submit the answer again but I only have 2 more tries, so I want to get some more advice before I do.

Your solution is correct.

Is there some special format you must adhere to when submitting solutions? For example, do you need to say exp(t) instead of e^t? Does it matter whether the terms come in a particular order, different from the one you wrote? For example, would $8 e^{8t} + 7t e^{t} - 6e^{t}$ or $8 e^{8t} - 6 e^{t} + 7 t e^{t}$ both be acceptable? Can you write $t e^{t}$ (or $t \exp(t)$), or must it be in the form $e^{t} \, t$ (or $\exp(t)\, t$), etc? I would certainly hope that the software would not be so picky, but who knows?

 

[Marco Lugo]

Your solution is correct.

Is there some special format you must adhere to when submitting solutions? For example, do you need to say exp(t) instead of e^t? Does it matter whether the terms come in a particular order, different from the one you wrote? For example, would $8 e^{8t} + 7t e^{t} - 6e^{t}$ or $8 e^{8t} - 6 e^{t} + 7 t e^{t}$ both be acceptable? Can you write $t e^{t}$ (or $t \exp(t)$), or must it be in the form $e^{t} \, t$ (or $\exp(t)\, t$), etc? I would certainly hope that the software would not be so picky, but who knows?

It worked! I had to use the exp(t) notation; which is weird because the software usually isn't picky and it would say if it wanted something expressed in a certain way.
Anyway thanks Mark44 and Ray Vickson for all you're help :D