Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
EXERCISES 1. Solve the following problems as indicated: b) a" = x. c) t²a" + 3tx' = + x = 0. d) tx" + 4x' + x = 0. t²x" - 7tx' +16x = 0. f) t²x" + 3tx'- 8x = 0, x(1) = 0, x'(1) = 2. g) t²x" + tx' = 0, x(1) = 0, x'(1) = 2. h) t²x" - tx' + 2x = 0, x(1) = 0, x' (1) = 1. 2. Solve the initial value problem x" + t²x' = 0, x(0) = 0, x'(0) = 1. Is this a Cauchy-Euler equation?

EXERCISES 1. Solve the following problems as indicated: b) a" = x. c) t²a" + 3tx' = + x = 0. d) tx" + 4x' + x = 0. t²x" - 7tx' +16x = 0. f) t²x" + 3tx'- 8x = 0, x(1) = 0, x'(1) = 2. g) t²x" + tx' = 0, x(1) = 0, x'(1) = 2. h) t²x" - tx' + 2x = 0, x(1) = 0, x' (1) = 1. 2. Solve the initial value problem x" + t²x' = 0, x(0) = 0, x'(0) = 1. Is this a Cauchy-Euler equation?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
Please do number 2
**Exercises on Second-Order Linear Equations** 1. Solve the following problems as indicated: a) \( x'' = -\frac{1}{t^2} x \). b) \( x'' = \frac{4}{t^2} x \). c) \( t^2 x'' + 3t x' = +x = 0 \). d) \( t^2 x'' + 4x' + \frac{2}{t} x = 0 \). e) \( t^2 x'' - 7t x' + 16x = 0 \). f) \( t^2 x'' + 3t x' - 8x = 0, \, x(1) = 0, \, x'(1) = 2 \). g) \( t^2 x'' + t x' = 0, \, x(1) = 0, \, x'(1) = 2 \). h) \( t^2 x'' - t x' + 2x = 0, \, x(1) = 0, \, x'(1) = 1 \). 2. Solve the initial value problem \( x'' + t^2 x' = 0, \, x(0) = 0, \, x'(0) = 1 \). Is this a Cauchy-Euler equation? 3. This exercise presents a method for solving a Cauchy-Euler equation using a change of the independent variable. Show that the transformation \( \tau = \ln t \) to a new independent variable \( \tau \) transforms the Cauchy-Euler equation \( at^2 x'' + bt x' + cx = 0 \) into a linear equation with constant coefficients. Use this method to solve Exercise 1a. 4. Find the general solution to the equation \( a(t) x'' + a'(t) x' = f(t) \). Your answer should be expressed in terms of integrals.
Transcribed Image Text:**Exercises on Second-Order Linear Equations** 1. Solve the following problems as indicated: a) \( x'' = -\frac{1}{t^2} x \). b) \( x'' = \frac{4}{t^2} x \). c) \( t^2 x'' + 3t x' = +x = 0 \). d) \( t^2 x'' + 4x' + \frac{2}{t} x = 0 \). e) \( t^2 x'' - 7t x' + 16x = 0 \). f) \( t^2 x'' + 3t x' - 8x = 0, \, x(1) = 0, \, x'(1) = 2 \). g) \( t^2 x'' + t x' = 0, \, x(1) = 0, \, x'(1) = 2 \). h) \( t^2 x'' - t x' + 2x = 0, \, x(1) = 0, \, x'(1) = 1 \). 2. Solve the initial value problem \( x'' + t^2 x' = 0, \, x(0) = 0, \, x'(0) = 1 \). Is this a Cauchy-Euler equation? 3. This exercise presents a method for solving a Cauchy-Euler equation using a change of the independent variable. Show that the transformation \( \tau = \ln t \) to a new independent variable \( \tau \) transforms the Cauchy-Euler equation \( at^2 x'' + bt x' + cx = 0 \) into a linear equation with constant coefficients. Use this method to solve Exercise 1a. 4. Find the general solution to the equation \( a(t) x'' + a'(t) x' = f(t) \). Your answer should be expressed in terms of integrals.
Expert Solution
Step 1

Given Data:

2.Let us consider the given equation is,

x''+t2x'=0 , x0=0 , x'0=1

steps

Step by step

Solved in 4 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS