# Solving Linear ODEs: Question & Answer

• physicsjock
In summary, the conversation discusses a question about solving a specific example problem and how to approach it. The person asking the question is confused about the given solution and asks for clarification. The expert suggests a method of solving the problem by substituting a given solution and notes a mistake in the person's calculation. The expert then gives a correction and offers a solution to the problem.
physicsjock
Hey,

I have been trying to work out how to solve this example question I found in a recommended text,

http://img715.imageshack.us/img715/5052/asdavm.jpg

Everywhere I've been reading starts with since y=eta(x) is a particular solution then y=eta(x) + u is a general solution, but the question doesn't state eta(x) as a particular solution, is that a problem?

Anyway in this question is the way to go simply subbing in y = eta + 1/si then noting that all the lone eta components cancel since they are a solution themselves.

But when I do that,

I end up with something like

$\frac{dx}{d\Psi }=\left[ p(x)+2q(x)\eta (x) \right]\Psi (x)+\frac{q(x)}{{{\Psi }^{2}}(x)}$

and I can't rearrange it to form the equation which si solves,

any idea on what I've done wrong?

Last edited by a moderator:
hey physicsjock!

where did that dx/dψ come from?

it should be -(1/ψ2)dψ/dx

(and then you'll be able to multiply throughout by ψ2)

ahah what a silly mistake I made!

Thanks heaps!

## What is a linear ODE?

A linear ODE (ordinary differential equation) is an equation that involves a function and its derivatives, where the function and derivatives are only raised to the first power and are multiplied by constants. It is called "linear" because the function and its derivatives are not multiplied together or raised to any powers other than 1.

## What are the steps for solving a linear ODE?

The steps for solving a linear ODE are as follows:

1. Identify the dependent variable and its derivatives in the equation.
2. Separate the equation into two sides, with the dependent variable terms on one side and the non-dependent variable terms on the other.
3. Integrate both sides of the equation.
4. Use initial conditions or boundary conditions to solve for any unknown constants.
5. Write the general solution, which includes all possible solutions to the equation.
6. Apply any additional conditions to find the particular solution, which is the solution that satisfies all given conditions.

## What are the common techniques for solving linear ODEs?

The most common techniques for solving linear ODEs include:

• Separation of variables
• Integrating factors
• Substitution
• Power series
• Variation of parameters
• Laplace transforms

## What are the applications of solving linear ODEs?

Solving linear ODEs has many practical applications in various fields such as physics, engineering, economics, and biology. Some common applications include modeling population growth, predicting the motion of objects, analyzing electrical circuits, and describing chemical reactions.

## What are the differences between linear and nonlinear ODEs?

The main difference between linear and nonlinear ODEs is that linear ODEs have a linear relationship between the function and its derivatives, while nonlinear ODEs do not. This means that the terms in a linear ODE are only raised to the first power and are multiplied by constants, whereas nonlinear ODEs can have higher powers and involve products of the function and its derivatives. Nonlinear ODEs are generally more difficult to solve than linear ODEs.

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