Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Solve the differential equation $f " (t) + 2 f' (t) + f (t) = 0$ and find all the real solutions of the differential equation.

The solution is $f (t) = e^{- t} (C_{1} + C_{2} t)$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: Definition of characteristic polynomial

Consider the linear differential operator

$T (f) = f^{(n)} + a_{n - 1} f^{(n - 1)} + a_{1} f' + a_{0} f$ .

The characteristic polynomial of is defined as

$P_{T} (λ) = λ^{n} + a_{n - 1} λ + . . . + a_{1} λ + a_{0}$ .

Step2: Determination of the solution

The characteristic polynomial of the operator as follows.

$T (f) = f " - 2 f' + f$

Then the characteristic polynomial is as follows.

$P_{T} (λ) = λ^{2} - 2 λ + 1$

Solve the characteristic polynomial and find the roots as follows.

$λ^{2} + 2 λ + 1 = 0 λ^{2} + λ + λ + 1 = 0 λ (λ + 1) + 1 (λ + 1) = 0 (λ + 1) (λ + 1) = 0$

Simplify further as follows.

$λ + 1 = 0 λ = - 1 λ + 1 = 0 λ = - 1$

Therefore, the roots of the characteristic polynomials are -1 and -1 .

Step3: Conclusion of the solution

Since, the roots of the characteristic equation is different real numbers.

The exponential functions $e^{- t}$ and ${te}^{- t}$ form a basis of the kernel of T .

Hence, they form a basis of the solution space of the homogenous differential equation is T(f) = 0 .

Thus, the general solution of the differential equation $f " (t) + 2 f' (t) = 0$ is $f (t) + e^{- t} (C_{1} + C_{2} t)$ .

Find Study Materials for

Create Study Materials

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Solve the differential equation $f " (t) + 2 f' (t) + f (t) = 0$ and find all the real solutions of the differential equation.

Step by Step Solution

Step1: Definition of characteristic polynomial

Step2: Determination of the solution

Step3: Conclusion of the solution

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Solve the differential equation f"(t)+2f'(t)+f(t)=0 and find all the real solutions of the differential equation.

Step by Step Solution

Step1: Definition of characteristic polynomial

Step2: Determination of the solution

Step3: Conclusion of the solution

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

Solve the differential equation $f " (t) + 2 f' (t) + f (t) = 0$ and find all the real solutions of the differential equation.