How Do You Solve Challenging Second-Order ODE Problems?

• tandoorichicken
In summary, the conversation discusses two problems: finding a particular solution using the method of undetermined coefficients and solving an IVP with initial conditions. The speaker is unsure of which coefficient expression to use in the first problem and is confused about how to account for the initial conditions in the second problem. They suggest using the initial conditions to fix the coefficient values.
tandoorichicken
There are a couple of problems on this week's homework assignment that are giving me trouble.

(1) Find a particular solution for $y'' + y = t^2$ by using the method of undetermined coefficients

Here I don't know which coefficient expression to use, for example if the term on the right side was e^t, I could sub y = Ae^t so that all 'e^t's would cancel out and I would be left with an expression for A.

(2) Solve the IVP: $y'' - 4y' +2y = e^{2t}$, homogenous initial conditions at t=0.

What I did was what I normally do for any first-order ODE. I separated the problem out into homogenous and particular parts.
$$y_h: y''-4y'+2y=0$$
$$s^2-4s+2=0\rightarrow s=2\pm\sqrt{2}$$ where s is a characteristic root. Therefore $y_h=c_1 e^{(2+\sqrt{2})t}+c_2 e^{(2-\sqrt{2})t}$
For the particular part I used undetermined coefficients and subbed y=Ae^2t and got an expression for A: 4A - 8A + 2A = 1, so A = -1/2, and $y_p = -\frac{1}{2}e^{2t}$
What I am confused about is what comes next.
Is the solution then just $y = y_p + y_h = c_1 e^{(2+\sqrt{2})t}+c_2 e^{(2-\sqrt{2})t} -\frac{1}{2}e^{2t}$ ? And how do I account for the initial conditions?

Initial conditions help you fix the coefficients values.They are arbitrary,unless there are initial conditions.

For the first,substitute

$$y_{p}(t)=C(x) \sin x$$

Daniel.

First of all, don't worry about feeling confused or struggling with certain problems in your homework. It's completely normal and part of the learning process. Just keep practicing and seeking help when needed.

To answer your first question about finding a particular solution for y'' + y = t^2, you can use the method of undetermined coefficients by considering the form of the right-hand side. Since t^2 is a polynomial of degree 2, you can assume a particular solution of the form y_p = At^2 + Bt + C. Then, substitute this into the equation and equate coefficients to find the values of A, B, and C.

For the second problem, you have correctly solved for the homogeneous solution and found the particular solution using undetermined coefficients. To account for the initial conditions, you can use the method of variation of parameters. This involves finding a particular solution in the form of y_p = u_1(t)y_1(t) + u_2(t)y_2(t), where y_1 and y_2 are the two linearly independent solutions of the homogeneous equation and u_1 and u_2 are functions to be determined. You can then use the initial conditions to solve for u_1 and u_2, and thus find the complete solution to the initial value problem.

I hope this helps clarify the next steps for solving these types of problems. Remember to always check your answers and seek help from your professor or classmates if needed. Keep up the good work!

1. What is a second-order ODE problem?

A second-order ordinary differential equation (ODE) is a mathematical equation that involves the second derivative of an unknown function. Solving these types of problems involves finding a function that satisfies the equation, subject to given initial or boundary conditions.

2. How are second-order ODE problems different from first-order ODE problems?

The main difference between first-order and second-order ODE problems is the number of derivatives involved. First-order ODEs only involve the first derivative of the unknown function, while second-order ODEs involve the second derivative. This often makes second-order ODEs more challenging to solve.

3. What are some real-world applications of second-order ODE problems?

Second-order ODE problems have many real-world applications, such as modeling the motion of a spring, analyzing electrical circuits, and predicting the growth of a population. They are also commonly used in physics, engineering, and economics to describe various physical phenomena.

4. What techniques are used to solve second-order ODE problems?

There are several techniques that can be used to solve second-order ODE problems, including separation of variables, substitution, and the method of undetermined coefficients. Other methods, such as power series and Laplace transforms, can also be used for more complex problems.

5. Are there any software programs or tools that can help solve second-order ODE problems?

Yes, there are many software programs and tools that can help with solving second-order ODE problems. Some popular options include MATLAB, Wolfram Alpha, and Maple. These programs use numerical methods to find approximate solutions to ODE problems, making it easier and faster to solve complex equations.

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