Explanation Given: The differential equation is given by, y ″ ( θ ) + 2 y ′ ( θ ) + 2 y ( θ ) = e − θ cos θ ... ( 1 ) Approach: 1. Find the characteristic equation by substitutions y ″ = r 2 , y ′ = r . 2. Find the roots of the characteristic equation. 3. If roots are real. Case-1 The general solution if r = r 1 = r 2 is, y ( t ) = c 1 e r t + c 2 t e r t Case-2 The general solution if r 1 ≠ r 2 is, y ( t ) = c 1 e r 1 t + c 2 e r 2 t 4. The general solution if roots are imaginary in form of r 1 , 2 = α ± β i is, y ( t ) = e a t ( c 1 cos β t + c 2 sin β t ) For each function in f i ( t ) in f ( t ) , determine the associated set of functions F i . 5. F i can be obtained as, The associate function if f i ( t ) = t m is, F = { 1 , t , t 2 , ... , t m } The associate function if f i ( t ) = t m e α t is, F = { e α t , e α t t , e α t t 2 , ... , e α t t m } The associate function if f i ( t ) = t m e α t cos β t or f i ( t ) = t m e α t sin β t is, F = { e α t cos β t , e α t t cos β t , e α t t 2 cos β t , ... , e α t t m cos β t , e α t sin β t , e α t t sin β t , e α t t 2 sin β t , ... , e α t t m sin β t } 6. The general solution if roots are imaginary in form of r 1 , 2 = α ± β i is, y ( t ) = e a t ( c 1 cos β t + c 2 sin β t ) The particular solution can be obtained by taking the linear combination of all functions in the associated sets. If any repeated function is present then it should appear only once in the particular solution. The general solution of the differential equation is given by, y = y c + y p ... ( 3 ) Here, y c is the complementary function and y p is the particular solution. Calculation: Consider the given differential equation. y ″ ( θ ) + 2 y ′ ( θ ) + 2 y ( θ ) = e − θ cos θ The auxiliary equation is written as follows r 2 + 2 r + 2 = 0 Solve the above equation for r . r 2 + 2 r + 2 = 0 r 2 + 2 r + 1 + 1 = 0 ( r + 1 ) 2 + 1 = 0 ( r + 1 ) 2 = − 1 Further simplify the above equation as, ( r + 1 ) = ± i r = − 1 ± i From the above equation the roots of the equation are, r 1 = − 1 + i r 2 = − 1 − i The complementary solution for the equation ( 1 ) using the roots of auxiliary equation is, y c = c 1 e − θ cos θ + c 2 e − θ sin θ Let the particular solution is, y p ( θ ) = θ e − θ ( A cos θ + B sin θ ) y p ( θ ) = e − θ ( A θ cos θ + B θ sin θ ) ... ( 4 ) Differentiate the above equation. y ′ p ( θ ) = d d θ e − θ ( A θ cos θ + B θ sin θ ) y ′ p ( θ ) = [ e − θ ( A ( − θ sin θ + cos θ ) + B ( θ cos θ + sin θ ) ) + ( A θ cos θ + B θ sin θ ) ( − e − θ ) ] y ′ p ( θ ) = e − θ [ ( − A − B ) θ sin θ + ( B − A ) θ cos θ + A cos θ + B sin θ ] Again differentiate the above equation.

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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 21E

To determine

To find:

The general solution to the given differential equation.

y″(θ)+2y′(θ)+2y(θ)=e−θcosθ

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.5, Problem 20E

Chapter 4.5, Problem 22E

Students have asked these similar questions

Consider the initial value problem y"+y'-12y= 0, y(0) = a, y'(0) = 4 Find the value of a so that the solution to the initial value problem approaches zero as too a =

Please refer below

Chapter 4 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

Ch. 4.1 - Verify that for b=0 and Fext(t)=0, equation (3)...Ch. 4.1 - If Fext(t)=0, equation (3) becomes my+by+ky=0. For...Ch. 4.1 - Show that if Fext(t)=0, m=1, k=9, and b=6, then...Ch. 4.1 - Prob. 4E Ch. 4.1 - Verify that the exponentially damped sinusoid...Ch. 4.1 - An external force F(t)=2cos2t is applied to a...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - Undamped oscillators that are driven at resonance...

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 general solution to the given differential equation. y ″ ( θ ) + 2 y ′ ( θ ) + 2 y ( θ ) = e − θ cos θ

Solution Summary: The author explains how to find the general solution to the given differential equation.

Concept explainers

Videos

To find:

The general solution to the given differential equation.

y″(θ)+2y′(θ)+2y(θ)=e−θcosθ

Want to see the full answer?

Chapter 4 Solutions

The general solution to the given differential equation. y ″ ( θ ) + 2 y ′ ( θ ) + 2 y ( θ ) = e − θ cos θ

Solution Summary: The author explains how to find the general solution to the given differential equation.

Concept explainers

Videos

To find: The general solution to the given differential equation. y″(θ)+2y′(θ)+2y(θ)=e−θcosθ

Want to see the full answer?

Chapter 4 Solutions

To find:

The general solution to the given differential equation.

y″(θ)+2y′(θ)+2y(θ)=e−θcosθ