# Science and engineering math: non-homogeneous differential equation

• chatterbug219
In summary: Figure out \frac{1}{(s-2)(s+2)(s^2+4)} = \frac{A}{s-2} + \frac{B}{s+2} + \frac{Cs + D}{s^2+4}, then multiply by the numerator.In summary, Homework Equations involve solving a differential equation with four roots, and taking the Laplace transform of the equation. Partial fractions can be used to simplify finding the roots, but the second equation is giving me trouble. I need to figure out how to do the partial fractions for it, and then I need to take the inverse Laplace of the second equation in order to find the solutions.
chatterbug219

## Homework Statement

Solve the differential equation
y(iv)(t) - 16y(t) = 30sint
subject to y(0) = 0, y'(0) = 2, y"(∏) = 0, y'"(∏) = -18

## Homework Equations

There is a Laplace transform table attached if needed :)

## The Attempt at a Solution

I tried making it homogeneous and then taking the Laplace. But I don't think that was right because wouldn't the Laplace of y(t) just be the definition of a Laplace ∫ y(t)e-stdt? So that's where I got confused

#### Attachments

• Laplace%20Transform%20Info.pdf
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hi chatterbug219!

just solve x4 - 16 = 0

(x2-4)(x2+4)=0
x=2

But how do I use that to solve the differential?

chatterbug219 said:
(x2-4)(x2+4)=0
x=2

there are four roots

haven't you covered how to solve polynomial differential equations?

chatterbug219 said:

## Homework Statement

Solve the differential equation
y(iv)(t) - 16y(t) = 30sint
subject to y(0) = 0, y'(0) = 2, y"(∏) = 0, y'"(∏) = -18

## Homework Equations

There is a Laplace transform table attached if needed :)

## The Attempt at a Solution

I tried making it homogeneous and then taking the Laplace. But I don't think that was right because wouldn't the Laplace of y(t) just be the definition of a Laplace ∫ y(t)e-stdt? So that's where I got confused

If you take the Laplace Transform of the DE, you need y''(0) = a and y'''(0) = b, which you are not given. However, you can solve for the transform Y(s) as a function of the unknown parameters a and b. Then you can get the solution y(t) in terms of a and b; then you can impose the boundary conditions at t = π to finally determine a and b.

RGV

So I did that and I got
Y= 30[1/(s2+1)(s4-16)] + (2s2+as+b)/(s4-16)
Do I do partial fractions for this now to get a and b? Then take the inverse Laplace?

chatterbug219 said:
So I did that and I got
Y= 30[1/(s2+1)(s4-16)] + (2s2+as+b)/(s4-16)
Do I do partial fractions for this now to get a and b? Then take the inverse Laplace?

Try it and see!

RGV

I've been trying to do the partial fractions for both...so far all I have is
1/(s2+1)(s4-16)
where I get
[As+B/(s2+1)] + [(Cs+D)/(s4-16)]
which simplifies to
[As(s4-16)]+[B(s4-16)]+[Cs(s2+1)]+[D(s2+1)]
and D=1 & C=-2/5...but I can't seem to get A or B to cancel so I can solve for them

the second one is (2s2+as+b)/(s4-16)
but I don't know how to do partial fractions for that

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chatterbug219 said:
I've been trying to do the partial fractions for both...so far all I have is
1/(s2+1)(s4-16)
where I get
[As+B/(s2+1)] + [(Cs+D)/(s4-16)]
which simplifies to
[As(s4-16)]+[B(s4-16)]+[Cs(s2+1)]+[D(s2+1)]
and D=1 & C=-2/5...but I can't seem to get A or B to cancel so I can solve for them

the second one is (2s2+as+b)/(s4-16)
but I don't know how to do partial fractions for that

Write $(s^4 - 16) = (s^2+4)(s^2-4) = (s-2)(s+2)(s^2+4)$ and go on from there. Alternatively, if you want to avoid quadratic denominators altogether, you could take complex roots and write $(s^4 - 16) = (s-2)(s+2)(s-2i)(s+2i),$ but that is not really necessary, since the inverse Laplace of $1/(s^2 + w^2)$ is tabulated.

RGV

So I got the partial fractions for the first one, but not the second one becaue the a and b are throwing me off...I don't know what to do with them in order to get the partial fractions of
[2s2+as+b]/[(s-2)(s+2)(s2+4)]

chatterbug219 said:
So I got the partial fractions for the first one, but not the second one becaue the a and b are throwing me off...I don't know what to do with them in order to get the partial fractions of
[2s2+as+b]/[(s-2)(s+2)(s2+4)]

Figure out $$\frac{1}{(s-2)(s+2)(s^2+4)} = \frac{A}{s-2} + \frac{B}{s+2} + \frac{Cs + D}{s^2+4},$$ then multiply by the numerator.

RGV

Okay...
-1/8(s2+4) - 1/32(s+2) + 1/32(s-2)

So do I multiply 2s2+as+b by -1/8(s2+4) - 1/32(s+2) + 1/32(s-2)? Is that you were saying?
But how do I take the inverse Laplace of 2s2+as+b? That's where I'm really confused

Last edited:

## 1. What is a non-homogeneous differential equation?

A non-homogeneous differential equation is a type of mathematical equation that involves a dependent variable, its derivatives, and an independent variable. It also includes a function that is not equal to zero, known as a forcing function, making it non-homogeneous.

## 2. How is a non-homogeneous differential equation different from a homogeneous one?

A homogeneous differential equation has a forcing function that is equal to zero, making it easier to solve. In contrast, a non-homogeneous differential equation requires additional techniques, such as variation of parameters or undetermined coefficients, to find a particular solution.

## 3. What is the role of initial conditions in solving non-homogeneous differential equations?

Initial conditions are important in solving non-homogeneous differential equations as they help determine the constants of integration needed to find the general solution. These conditions provide specific values for the dependent variable and its derivatives at a given point, which can be used to solve for the constants.

## 4. How are non-homogeneous differential equations used in science and engineering?

Non-homogeneous differential equations are used in many fields of science and engineering to model real-world phenomena. They can be used to describe the movement of objects, the growth of populations, the spread of diseases, and many other natural and physical processes.

## 5. What are some common techniques for solving non-homogeneous differential equations?

Some common techniques for solving non-homogeneous differential equations include the method of undetermined coefficients, variation of parameters, and Laplace transforms. Each method has its advantages and may be more suitable for certain types of non-homogeneous equations.

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