Calculus II - Solving Second Order Differential Equation

[GreenPrint]

Homework Statement

Which of the following functions are soltuions of the differntial equation y''+y=sin(x)?
a) y=cos(x) b) y=sin(x) c) y=1/2*xcos(x) d) y=1/2*x*sin(x)

Homework Equations

The Attempt at a Solution

I'm kind of lost on how to solve this problem. I don't think this is a standard calculus II problem but it was on a packet that we were asked to solve.

I learned from the internet that you can solve a second differential order equation of form
Af''(x) + Bf'(x) + Cf(x) = 0
By setting f'(x) as r_1, f''(x) as r_2, f(x) as 1
then
Ar^2 + Br + C = 0
and plugging into
y(x) = c_1 e^(r_1*x) + c_2*e^(r_2*x)
were c_1 and c_2 are just some constants that can be solved for if your given initial conditions

I'm unsure how to solve this problem because it's of a different form of
f''(x) + f(x) = sin(x)
I'm unsure what to do about the sin(x) term in this case and not sure what to do sense I don't have initial condition and how to come up with the constant terms as a result...

Thanks for any Help!

 

[Hootenanny]

This type of equation is known as an Inhomogeneous second order ordinary differential equation with constant coefficients. It is the presence of the $\sin x$ on the RHS that makes it "inhomogeneous".

Have you met this type of equation before? If not, take a look here: http://www.cse.salford.ac.uk/profil...cond-order-differential-equations-inhomog.pdf

 

[GreenPrint]

Thanks for your response. No I have never seen type of equation before. Is it a standard calculus II topic? That pdf file looks very useful, I'll see if I can solve the problem after looking over it.

 

[Hootenanny]

Thanks for your response. No I have never seen type of equation before. Is it a standard calculus II topic? That pdf file looks very useful, I'll see if I can solve the problem after looking over it.

I am not familiar with the US education system, so I can't comment. However, over here in the UK, this is first year undergraduate level.

 

[mstud]

Homework Statement

Which of the following functions are soltuions of the differential equation y''+y=sin(x)?
a) y=cos(x) b) y=sin(x) c) y=1/2*xcos(x) d) y=1/2*x*sin(x)

Thanks for any Help!

You can also, when checking for solutions, think in the other direction, which makes the calculation quite simple.

find y'' of the suggested y=...

Calculate y''+y and see if this equals sin(x)

I think you'll find that's far easier than solving the 2nd order differential equation.

When I learned about diff eqs this spring we started approaching possible solutions this way, later we learn't how to solve such differential equations.

 

[dynamicsolo]

You are not being asked to solve the differential equation (inhomogeneous second-order ordinary differential equations are not a Calculus II topic), but to "verify" a solution. Which choice works in the equation?

 

[GreenPrint]

I'm actually getting none of them

a) y=cos(x)
y'=-sin(x)
y''=-cos(x)
y''+y=-cos(x)+cos(x)=0

b)y=sin(x)
y'=cos(x)
y''=-sin(x)
y''+y=-sin(x)+sin(x)=0

c)y=(xcos(x))/2
y'=(-xsin(x))/2+cos(x)/2
y''=-(xcos(x))/2-sin(x)
y''+y=(-xcos(x))/2-sin(x)+(xcos(x))/2=-sin(x)

d)y=(xsin(x))/2
y'=sin(x)/2+(xcos(x))/2
y''=cos(x)/2+cos(x)/2-(xsin(x))/2=cos(x)-(xsin(x))/2
y''+y=cos(x)-(xsin(x))/2+(xsin(x))/=cos(x)

Is the problem missed up or am I doing something wrong because I'm getting that all the answers are wrong?

 

[mstud]

I can't find something wrong there...

 

[Mark44]

As you show the problem, none of the given functions is a solution. It would be a good idea to check that you wrote them down correctly, particularly the ones for parts c and d. If you missed a sign, that would affect your answer.

 

[Hootenanny]

You are not being asked to solve the differential equation (inhomogeneous second-order ordinary differential equations are not a Calculus II topic), but to "verify" a solution. Which choice works in the equation?

None of them do, which is why I suggested that the OP look into how to solve the problem himself. I wasn't sure whether that was part of the exercise or not.

 

[dynamicsolo]

None of them do, which is why I suggested that the OP look into how to solve the problem himself. I wasn't sure whether that was part of the exercise or not.

If it's a multiple-choice question for practice, finding the error is probably not part of the exercise.

I guess I want to ask whether this set of problems came from a published source or from an instructor. It wouldn't be the first time I'd seen someone make up a bunch of problems and not check for typoes or whether a problem could even be solved as stated. I suspect the answer was intended to be (c), but a minus sign was omitted on the inhomogeneous term or on the solution. [I agree that there are no errors in GreenPrint's calculations.]

 

[GreenPrint]

It actually came from my professor, I guess the problem is messed up, I'm glad I didn't do something wrong =). It's strange, the difference between education systems, it's something taught in first year undergraduate math but not in the US apparently... I guess they have better education systems elsewhere lol, that doesn't surprise me

 

[Hootenanny]

If it's a multiple-choice question for practice, finding the error is probably not part of the exercise.

I guess I want to ask whether this set of problems came from a published source or from an instructor. It wouldn't be the first time I'd seen someone make up a bunch of problems and not check for typoes or whether a problem could even be solved as stated. I suspect the answer was intended to be (c), but a minus sign was omitted on the inhomogeneous term or on the solution. [I agree that there are no errors in GreenPrint's calculations.]

Fair point. I just didn't want to give the answer away if this wasn't the case

 

[Mark44]

Solving the differential equation is probably too advanced for a Calc II course, but verifying that a given function is a solution of a differential equation is appropriate, IMO. For problems like this, all you're doing is finding out whether the given function and it's relevant derivatives make the differential equation identically true.

 

[vela]

It actually came from my professor, I guess the problem is messed up, I'm glad I didn't do something wrong =). It's strange, the difference between education systems, it's something taught in first year undergraduate math but not in the US apparently... I guess they have better education systems elsewhere lol, that doesn't surprise me

I seem to recall the variation of parameters method was covered in the second semester of calculus when I took it. You can use that method to solve this equation.