Explanation Formula used: General solution of the Cauchy-Euler DE. 1 Consider the DE and find the complementary solution. 2 Consider that the complete solution of DE is y ( x ) = u v . 3 Find v using the variable separation and multiplication of integration factor. 4 Find the general solution and dictate. Calculation: Consider the given Cauchy-Euler’s DE x 4 y ″ + x 3 y ′ − 4 x 2 y = 1 . Given that one of the solution of the DE is y 1 = x 2 . Take the common factor x 4 out of the DE and modify the DE as shown below. y ″ + 1 x y ′ − 4 x 2 y = 1 x 4 ⋯ ( 1 ) Compare the equation ( 1 ) with standard form y ″ + P y ′ + Q y = R . Therefore, P = 1 x , Q = − 4 x 2 and R = 1 x 4 . Consider that the given solution as y 1 = u … ( 2 ) . Now, let the complete the solution of the DE be y = u v . Indicate the DE ( 1 ) in the terms of the function v . Therefore, v ″ + ( P + 2 u d u d x ) v ′ = R u ⋯ ( 3 ) Substitute P = 1 x , Q = − 4 x 2 and R = 1 x 4 into v ″ + ( P + 2 u d u d x ) v ′ = R u . Therefore, v ″ + ( 1 x + 2 x 2 d u d x ) v ′ = 1 x 6 Consider that v ′ = q and v ″ = q ′ . Therefore, the reduced from of the DE v ″ + ( 1 x + 2 x 2 d u d x ) v ′ = 1 x 6 is q ′ + ( 5 x q ) = 1 x 6 [ Since u = x 2 and d u d x = 2 x ] . Compare the DE q ′ + ( 5 x q ) = 1 x 6 to standard form d y d x + P y = Q . Therefore, find the integral I F = e ∫ P d x = e ∫ 5 x d x = e 5 ln x [ Ignore the integral constant ] = x 5 The solution of the q ′ + ( 5 x q ) = 1 x 6 is q ⋅ I F = ∫ Q ⋅ I F d x + c 1

Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th

11th Edition

ISBN: 9781305965737

Author: Dennis G. Zill

Publisher: Brooks Cole

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Chapter 4.6, Problem 36E

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The general solution of the differential equation x4y″+x3y′−4x2y=1.

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Chapter 4.6, Problem 35E

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Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th

Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - Given that y=c1+c2x2 is a two-parameter family of...Ch. 4.1 - Find two members of the family of solutions in...Ch. 4.1 - Given that x(t) = c1 cos t + c2 sin t is the...Ch. 4.1 - Use the general solution of x + 2x = 0 given in...Ch. 4.1 - In Problems 9 and 10 find an interval centered...Ch. 4.1 - In Problems 9 and 10 find an interval centered...

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The general solution of the differential equation x 4 y ″ + x 3 y ′ − 4 x 2 y = 1 .

Solution Summary: The author explains the general solution of the Cauchy-Euler's DE.

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