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2. Find the ger 3. Find the general solution of the equation t2x" - 3tx' + 3x = 4t7. r(t) be independent solutions of the m (+) and a =

2. Find the ger 3. Find the general solution of the equation t2x" - 3tx' + 3x = 4t7. r(t) be independent solutions of the m (+) and a =

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
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124 EXERCISES 1. Use the variation of parameters formula to find a particular solution to the following equations. a) x" + x = tant. b) x" - x = tet. c) x" - x = 1. 2. Second-Order Linear Equations 2. Find the general solution of the equation x"+x' = a, where a is a const 3. Find the general solution of the equation t²x" - 3tx' + 3x = 4t7. 4. Let x₁ = x1(t) and x2 = x2(t) be independent solutions of the equation "+p(t)x' + q(t)x = 0 on an interval I and let W(t) E Wronskian of X1 and x2. d) t²x"-2x = t³. e) x" + x = 1 f) x" - 2x + x = 1et. a) By direct differentiation show that the Wronskian satisfies the ential equation W'(t) = -p(t)W(t). = b) Solve this equation for W(t) and show that either W(t) 0 te I, or W(t) is never zero on I. 5. Let Lx = x" + px' + cx, where r expression. Suppose x o(t) is 0, '(0) = 1. Show that a par xp (t) = f(ts)f(s)ds, wh c) If t = a and t = b are two adjacent zeros of x₁(t), i.e., x₁(a) = 0, show that x2(t) must have a zero between a and b. Hint: U (b). = constant, to Lo ion to Lx = constant. be C
Transcribed Image Text:124 EXERCISES 1. Use the variation of parameters formula to find a particular solution to the following equations. a) x" + x = tant. b) x" - x = tet. c) x" - x = 1. 2. Second-Order Linear Equations 2. Find the general solution of the equation x"+x' = a, where a is a const 3. Find the general solution of the equation t²x" - 3tx' + 3x = 4t7. 4. Let x₁ = x1(t) and x2 = x2(t) be independent solutions of the equation "+p(t)x' + q(t)x = 0 on an interval I and let W(t) E Wronskian of X1 and x2. d) t²x"-2x = t³. e) x" + x = 1 f) x" - 2x + x = 1et. a) By direct differentiation show that the Wronskian satisfies the ential equation W'(t) = -p(t)W(t). = b) Solve this equation for W(t) and show that either W(t) 0 te I, or W(t) is never zero on I. 5. Let Lx = x" + px' + cx, where r expression. Suppose x o(t) is 0, '(0) = 1. Show that a par xp (t) = f(ts)f(s)ds, wh c) If t = a and t = b are two adjacent zeros of x₁(t), i.e., x₁(a) = 0, show that x2(t) must have a zero between a and b. Hint: U (b). = constant, to Lo ion to Lx = constant. be C
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