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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?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 94E
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120 EXERCISES 1. Solve the following problems as indicated: a) a" = -x. b) a" = x. c) t²x" + 3tx' = + x = 0. d) tx" + 4x' + x = 0. e) t²x" - 7tx' + 16x = 0. 2. Second-Order Linear Equations 2.4.2 Variation of Parameter f) t²x" + 3tx' - 8x = 0, x(1) = 0, x'(1) = 2. There is a general formula, call particular solution to a nonb g) t²x" + tx' = 0, 2. Solve the initial value problem x" + t²x' = 0, x(0) = 0, x'(0) = 1. Is this a Cauchy-Euler equation? It requires knowledge x(1) = 0, x'(1) = 2. 3. This exercise presents a method for solving a Cauchy-Euler equation using a change of the independent variable. Show that the transformation 7 = Int to a new independent variable T transforms the Cauchy-Euler equation at²x"+btx' + cx = 0 into an linear equation with constant coefficients. Use this method to solve Exercise la. h) t²x" - tx' + 2x = 0, x(1) = 0, x' (1) = 1. 4. Find the general solution to the equation a(t)x" + a' (t)x' = f(t). You answer should be expressed in terms of integrals. n of ear equat q(t)x= f' undamen parar
Transcribed Image Text:120 EXERCISES 1. Solve the following problems as indicated: a) a" = -x. b) a" = x. c) t²x" + 3tx' = + x = 0. d) tx" + 4x' + x = 0. e) t²x" - 7tx' + 16x = 0. 2. Second-Order Linear Equations 2.4.2 Variation of Parameter f) t²x" + 3tx' - 8x = 0, x(1) = 0, x'(1) = 2. There is a general formula, call particular solution to a nonb g) t²x" + tx' = 0, 2. Solve the initial value problem x" + t²x' = 0, x(0) = 0, x'(0) = 1. Is this a Cauchy-Euler equation? It requires knowledge x(1) = 0, x'(1) = 2. 3. This exercise presents a method for solving a Cauchy-Euler equation using a change of the independent variable. Show that the transformation 7 = Int to a new independent variable T transforms the Cauchy-Euler equation at²x"+btx' + cx = 0 into an linear equation with constant coefficients. Use this method to solve Exercise la. h) t²x" - tx' + 2x = 0, x(1) = 0, x' (1) = 1. 4. Find the general solution to the equation a(t)x" + a' (t)x' = f(t). You answer should be expressed in terms of integrals. n of ear equat q(t)x= f' undamen parar
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