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Use Cramer's Rule to solve this system for u and u½ and show that the solutions u1 and uz are given by - [ V½f(x) -dx U1 W Yıf (x) -dx W U2 Y1 Y2 ly,' vis the Wronskian. where the determinant W =

Use Cramer's Rule to solve this system for u and u½ and show that the solutions u1 and uz are given by - [ V½f(x) -dx U1 W Yıf (x) -dx W U2 Y1 Y2 ly,' vis the Wronskian. where the determinant W =

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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Variation of parameters for a single ODE starts with the guess Yp (x) = u1 (x)y1 (x) + u2(x)y2(x), where y, (x) and y2 (x) are the known linearly independent solutions to the homogenous problem and u, (x) and u2(x) are unknown functions of x. For simplicity, the dependence on x will not be explicitly noted in the remainder of this problem, i.e. u1 = u1 (x), etc. Substituting this guess for y, into the standard form y" + p(x)y' + q(x)y = f(x) and making simplifications then leads to this system of two algebraic equations: uiy1 + u½y2 = 0 uiyi + uży2 = f (x) Use Cramer's Rule to solve this system for u and uz and show that the solutions u and uz are given by Y2f (x) 2dx W U1 Yıf (x) Uz W Y2 is the Wronskian. Y2 where the determinant W =
Transcribed Image Text:Variation of parameters for a single ODE starts with the guess Yp (x) = u1 (x)y1 (x) + u2(x)y2(x), where y, (x) and y2 (x) are the known linearly independent solutions to the homogenous problem and u, (x) and u2(x) are unknown functions of x. For simplicity, the dependence on x will not be explicitly noted in the remainder of this problem, i.e. u1 = u1 (x), etc. Substituting this guess for y, into the standard form y" + p(x)y' + q(x)y = f(x) and making simplifications then leads to this system of two algebraic equations: uiy1 + u½y2 = 0 uiyi + uży2 = f (x) Use Cramer's Rule to solve this system for u and uz and show that the solutions u and uz are given by Y2f (x) 2dx W U1 Yıf (x) Uz W Y2 is the Wronskian. Y2 where the determinant W =
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