Explanation Calculation: For the differential equation y ' ' + 2 y ' + y = x 2 e − x , the corresponding homogeneous differential equation is y ' ' + 2 y ' + y = 0 The corresponding auxiliary equation is: m 2 + 2 m + 1 = 0 m 2 + 2 m + 1 = 0 ( m + 1 ) 2 = 0 m 1 = m 2 = − 1 Therefore, the complementary function is: y c = c 1 e − x + c 2 x e − x Since, x 2 e − x is annihilated by the differential operator ( D + 1 ) 3 . Therefore, apply the differential operator ( D + 1 ) 3 to both sides of the equation y ' ' + 2 y ' + y = x 2 e − x ( D + 1 ) 3 ( D 2 + 2 D + 1 ) y = ( D + 1 ) 3 ( x 2 e − x ) ( D + 1 ) 3 ( D 2 + 2 D + 1 ) y = 0 The corresponding auxiliary equation is: ( m + 1 ) 3 ( m 2 + 2 m + 1 ) = 0 ( m + 1 ) 5 = 0 The roots are: m 1 = m 2 = m 3 = m 4 = m 5 = − 1 Therefore, its general solution is y = c 1 e − x + c 2 x e − x + c 3 x 3 e − x + c 4 x 4 e − x The particular solution y p of the given differential equation consists of the terms remaining in the above general equation which are not present in the complementary function of the original equation. Let y p = A x 3 e − x + B x 4 e − x

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

See similar textbooks

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 differentia…

Videos

Question

Chapter 4.5, Problem 52E

To determine

The solution of the differential equation y''+2y'+y=x2e−x by undetermined coefficients.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.5, Problem 51E

Chapter 4.5, Problem 53E

Students have asked these similar questions

T3.2: Prove that if the Graceful Tree Conjecture (every tree has a graceful labeling) is true and T' is a tree with m edges, then K2, decomposes into 2m - 1 copies of T. Hint - Delete a leaf to get 7" and apply the decomposition of K2(m-1)+1 = K2m-1 into T'. Then explain how the decomposition allows the pendant edge to be added to a new vertex to obtain a decomposition of K2m into copies of T.

Use the matrix tree theorem to determine the number of spanning trees of the graphs Kr∨sK1.These are the graphs formed by by adding all edges between a complete graph on r vertices and atrivial graph (no edges) on s vertices.

The maximum capacity spanning tree problem is as follows for a given graph G = (V, E) withcapacities c(uv) on the edges. The capacity of a tree T is defined as the minimum capacity of anedge in T. The maximum capacity spanning tree problem is to determine the maximum capacity ofa spanning tree.(i) Describe how to modify the input graph to find a maximum weight spanning tree making use ofa minimum weight spanning tree algorithm.(ii) Show that a maximum (weight) spanning tree is also a maximum capacity spanning tree.(iii) Is the converse of part (ii) true? That is, is it true that a maximum capacity spanning tree is alsoa maximum spanning tree? Either give counterexamples (of all sizes) or a proof.(iv) Prove the following max-min result. The maximum capacity of a spanning tree is equal to theminimum bottleneck value of a cut. For a subset U ⊆ V , the cut [U, V − U] is the set of edgesbetween U and V − U. The bottleneck value of a cut [U, V − U] is the largest capacity among theedges of…

Chapter 4 Solutions

Differential Equations with Boundary-Value Problems

Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - Given that y=c1+c2x2 is a two-parameter family of...Ch. 4.1 - Find two members of the family of solutions in...Ch. 4.1 - Prob. 7E Ch. 4.1 - Use the general solution of x + 2x = 0 given in...Ch. 4.1 - In Problems 9 and 10 find an interval centered...Ch. 4.1 - In Problems 9 and 10 find an interval centered...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The solution of the differential equation y ' ' + 2 y ' + y = x 2 e − x by undetermined coefficients.

Solution Summary: The author explains the solution of the differential equation y''+2y's +y=x2'e'-x by undetermined coefficients.

Concept explainers

Videos

The solution of the differential equation y''+2y'+y=x2e−x by undetermined coefficients.

Want to see the full answer?

Chapter 4 Solutions