# ODE Theory: General Solution to y'' + 4y' + 4y = 0

• smerhej
In summary, the problem required finding the general solution of the equation y'' + 4y' + 4y = 0 using reduction of order. The characteristic equation had a double root of -2, which prompted the use of reduction of order to obtain a second equation. However, the given solution only included c1e^-2t + c2e^-2t, leading to confusion about the proper method. Upon attempting reduction of order, a solution of y = u(t)e^-2t was found, resulting in u(t) = A + Bt. This leads to a general solution of y = Ae^-2t + Bte^-2t.
smerhej

## Homework Statement

We have y'' + 4y' + 4y = 0 ; find the general solution.

## Homework Equations

Reduction of Order.

## The Attempt at a Solution

So when determining the roots of the characteristic equation, -2 was a double root, and therefore we can't simply have c1e-2t + c2e-2t. So I thought I would use reduction of order to get a second equation. However in the solution, they just left it c1e-2t + c2e-2t and I'm wondering if what I was taught to do in the case of non distinct roots was wrong, or if the solution is wrong.

Last edited:
The solution appears to be wrong.

y = c1*exp(-2t) + c2 * t * exp(-2t)

Please show exactly what you did in your attempted reduction of order. When I try a solution of the form $y= u(t)e^{-2t}$, I get $u(t)= A+ Bt$ giving $y= Ae^{-2t}+ Bte^{-2t}$ as general solution.

## 1. What is an ODE?

An ODE, or ordinary differential equation, is a type of mathematical equation that involves a function and its derivatives. It is commonly used to model systems and phenomena in fields such as physics, engineering, and biology.

## 2. What is the general solution to y'' + 4y' + 4y = 0?

The general solution to this ODE is y = C1e^(-2x) + C2xe^(-2x), where C1 and C2 are arbitrary constants. This solution can be found by solving the characteristic equation r^2 + 4r + 4 = 0 and using the method of undetermined coefficients.

## 3. How do you solve an ODE using the method of undetermined coefficients?

The method of undetermined coefficients involves guessing a solution to the ODE that includes arbitrary constants, plugging it into the equation, and solving for the constants using algebraic manipulation. The guessed solution must include all terms present in the ODE, as well as any necessary derivatives.

## 4. What are the steps for solving a second-order linear ODE?

The general steps for solving a second-order linear ODE are as follows: 1. Write the ODE in standard form, with the highest derivative term written first.2. Find the roots of the characteristic equation.3. Use the method of undetermined coefficients to guess a solution and find the constants.4. Add the general solutions to the homogeneous and particular solutions to get the general solution to the ODE.

## 5. What are some applications of ODEs in science?

ODEs have many applications in science, including modeling the motion of objects under the influence of forces (such as in classical mechanics), describing the growth and decay of populations (such as in biology), and predicting the behavior of electrical circuits (such as in electrical engineering). They are also used in other fields such as economics, chemistry, and neuroscience.

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