# Help with nonhomogeneous linear equation

• Song
In summary, the conversation discusses a boundary-value problem y''+vy=0 with specific conditions y(0)=0 and y(L)=0. It is shown that for v=0 and v<0, the only solution is y=0. For v>0, the solution is y=c1 cos sqrt(v)x+c2 sin sqrt(v)x. The conversation also mentions a nontrivial solution for v>0 and provides an incorrect attempt at solving it.
Song
18.1
#33
Let L be a nonzero real #
(a) Show that the boundary-value problem y''+vy=0, y(0)=0, y(L)=0, has only the trivial solution y=0 for the cases v=0 and v<0.

I get (a), but I don't know how to do (b)

(b) For the cases v>0, find the values of v for which this problem has a nontrivial solution and give the corresponding solution.

Basically, I get the y=0 for (b) again. And this is wrong, but I don't know what is wrong.
r^2+v=0
r=+-sqrt(v)i because v>0.
Then y=c1 cos sqrt(v)x+c2 sin sqrt(v)x
From y(0)=0 & y(L)=0, I get the same answer as (a)...

You have:
$$y(x)=c_1 \cos (\sqrt{v}x) + c_2 \sin (\sqrt{v}x)$$
and
$$y(0)=0$$

Which very conveniently leads to something.

From there, you can use
$$y(L)=0$$
to figure out something about $\sqrt{v}$

I got c1=0, and c2=0
But they're not right..

...

## What is a nonhomogeneous linear equation?

A nonhomogeneous linear equation is an equation in which the terms involve both a dependent variable and its derivatives, as well as a constant term. It can be written in the form: y′′ + p(x)y′ + q(x)y = g(x), where p(x) and q(x) are functions of x and g(x) is a function of x only.

## How can I solve a nonhomogeneous linear equation?

To solve a nonhomogeneous linear equation, you can use the method of undetermined coefficients or the method of variation of parameters. In the method of undetermined coefficients, you assume a particular form for y(x) and solve for the coefficients. In the method of variation of parameters, you assume a particular form for the solution and then use a variation of the constants to solve for the coefficients.

## What are the applications of nonhomogeneous linear equations?

Nonhomogeneous linear equations have many applications in physics, engineering, and economics. They can be used to model and solve problems involving motion, heat transfer, population growth, and more. They are also used in signal processing and control systems.

## Can a nonhomogeneous linear equation have more than one solution?

Yes, a nonhomogeneous linear equation can have infinitely many solutions. This is because the general solution of a nonhomogeneous linear equation includes both the complementary function (the solution to the corresponding homogeneous equation) and a particular solution (a specific solution to the nonhomogeneous equation). Therefore, the general solution can be written as y(x) = yc(x) + yp(x), where yc(x) and yp(x) are the complementary function and particular solution, respectively.

## What is the difference between a homogeneous and nonhomogeneous linear equation?

The main difference between a homogeneous and nonhomogeneous linear equation is that the terms in a homogeneous equation do not include a constant term. Therefore, the general solution of a homogeneous linear equation only includes the complementary function, while the general solution of a nonhomogeneous linear equation includes both the complementary function and a particular solution.

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