# Undetermined Coefficients: Finding a Particular SolutionIn this section, we will use the method of undetermined coefficients to determine a particular solution to the given second-order linear nonhomogeneous differential equation:\[ y'' - 4y' + 2y = 4t + 7 \]The method of undetermined coefficients involves making an educated guess about the form of the particular solution based on the right-hand side of the equation, then determining the unknown coefficients by substituting our guess into the differential equation.Below is the differential equation and the box where you need to input the particular solution:\[ y'' - 4y' + 2y = 4t + 7 \]\[ y_p(t) = \boxed{\phantom{answer}} \]To solve this step-by-step:1. Identify the form of the particular solution $ y_p(t) $.2. Substitute $ y_p(t) $ into the differential equation.3. Solve for the unknown coefficients.Let's begin solving this equation by guessing the form of the particular solution.Given right-hand side: $ 4t + 7 $Suggested form: $ y_p(t) = At + B $Substitute $ y_p(t) = At + B $ into the differential equation and solve for $ A $ and $ B $. Once solved, you will enter the resulting particular solution in the box provided.Happy learning!

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se undetermined coefficients to find the particular solution to "- 4y' + 2y = 4t + 7 (t) =

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

# Undetermined Coefficients: Finding a Particular Solution

In this section, we will use the method of undetermined coefficients to determine a particular solution to the given second-order linear nonhomogeneous differential equation:

\[ y'' - 4y' + 2y = 4t + 7 \]

The method of undetermined coefficients involves making an educated guess about the form of the particular solution based on the right-hand side of the equation, then determining the unknown coefficients by substituting our guess into the differential equation.

Below is the differential equation and the box where you need to input the particular solution:

\[ y'' - 4y' + 2y = 4t + 7 \]

\[ y_p(t) = \boxed{\phantom{answer}} \]

To solve this step-by-step:
1. Identify the form of the particular solution $ y_p(t) $.
2. Substitute $ y_p(t) $ into the differential equation.
3. Solve for the unknown coefficients.

Let's begin solving this equation by guessing the form of the particular solution.

Given right-hand side: $ 4t + 7 $

Suggested form: $ y_p(t) = At + B $

Substitute $ y_p(t) = At + B $ into the differential equation and solve for $ A $ and $ B $. Once solved, you will enter the resulting particular solution in the box provided.

Happy learning!

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