Variation of parameters for a single ODE starts with the guessYp (x) = u1 (x)y1 (x) + u2(x)y2(x), where y, (x) and y2 (x) are the known linearlyindependent solutions to the homogenous problem and u, (x) and u2(x) areunknown functions of x. For simplicity, the dependence on x will not be explicitlynoted in the remainder of this problem, i.e. u1 = u1 (x), etc. Substituting this guessfor y, into the standard form y" + p(x)y' + q(x)y = f(x) and makingsimplifications then leads to this system of two algebraic equations:uiy1 + u½y2 = 0uiyi + uży2 = f (x)Use Cramer's Rule to solve this system for u and uz and show that the solutions uand uz are given byY2f (x)2dxWU1Yıf (x)UzWY2is the Wronskian.Y2where the determinant W =

Cramer's rule will give us the unique solution of a system of equations if it exists. However, if…

Answered: Use Cramer's Rule to solve this system…

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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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