# Find sol'n of nonhomogenous differential equation

• accountkiller
In summary: Wait, since y = yp + c1y1 + c2y2 and I know what yp and y1 and y2 are, I can just plug it in, use the initial conditions to solve for c1 and c2, and I get the correct answer, which is y = 2cos(x) - 5sin(x) + 3x.
accountkiller

## Homework Statement

A nonhomogeneous differential equation, a complimentary solution yc, and a particular solution yp are given. Find a solution satisfying the initial conditions.

y'' + y = 3x, y(0) = 2, y'(0) = -2, yc = c1cos(x) + c2sin(x), yp = 3x.

## Homework Equations

y = yp + c1y1 + ... + cnyn

## The Attempt at a Solution

So I first tried solving for the associated homogenous equation y'' + y = 0.
Guessing that y = erx, y' = rerx and y'' = r2erx, so
y'' + y = r2erx + erx = erx(r2+1) = 0, so
r2 + 1 = 0...
but that gives me a square root of a negative number for r?

For complex numbers, all I have in my notes from lecture is this example:
y1 = er1x and y2 = er2x with r1 = A + Bi. Solve the DE.
y'' - 2Ay' + (A2 + B2)y = 0.

I'm not sure how to use that.

Wait, since y = yp + c1y1 + c2y2 and I know what yp and y1 and y2 are, I can just plug it in, use the initial conditions to solve for c1 and c2, and I get the correct answer, which is y = 2cos(x) - 5sin(x) + 3x.

I don't have to do anything with the roots?
In a similar example problem the teacher did, he used roots (although not complex roots like this one has).

Wait, since y = yp + c1y1 + c2y2 and I know what yp and y1 and y2 are, I can just plug it in, use the initial conditions to solve for c1 and c2, and I get the correct answer, which is y = 2cos(x) - 5sin(x) + 3x.
Right - they have already told you what the solutions are, so there's no point in doing the work to find them.
I don't have to do anything with the roots?
In a similar example problem the teacher did, he used roots (although not complex roots like this one has).

## 1. What is a nonhomogenous differential equation?

A nonhomogenous differential equation is a type of mathematical equation that involves an unknown function and its derivatives, along with additional terms that do not depend on the unknown function. These additional terms make the equation nonhomogenous, as opposed to a homogenous differential equation where all terms involve the unknown function.

## 2. How do you solve a nonhomogenous differential equation?

There are several methods for solving a nonhomogenous differential equation, including the method of undetermined coefficients and the method of variation of parameters. Both of these methods involve finding a particular solution to the equation and combining it with the general solution of the corresponding homogenous equation.

## 3. What is the difference between a particular and general solution?

A particular solution is a specific solution to a differential equation, while a general solution contains all possible solutions to the equation. The general solution of a nonhomogenous differential equation will also include a particular solution, which is necessary to account for the additional terms in the equation.

## 4. Can a nonhomogenous differential equation have multiple solutions?

Yes, a nonhomogenous differential equation can have multiple solutions. This is because the general solution of the equation will contain a constant of integration, which can take on different values and result in different particular solutions. Additionally, some nonhomogenous equations may have multiple particular solutions due to the complexity of the equation.

## 5. What are the applications of solving nonhomogenous differential equations?

Nonhomogenous differential equations are used to model many real-world phenomena, such as population growth, chemical reactions, and electrical circuits. By solving these equations, scientists and engineers can make predictions and understand the behavior of these systems. Nonhomogenous differential equations are also important in the field of physics, where they are used to describe the motion of objects under the influence of external forces.

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