Consider the following initial value problem: (1) x2y" + 7xy' +9y = 27 In x , x > 0 y(1) = 1, y'(1) = -4 a. i. Solve the corresponding homogeneous equation as a Cauchy-Euler equation. ii. Determine yp by the method of variation of parameters. b. i. By using the following change of variables: x = et Show that Eq. (1) becomes a non-homogeneous equation with constant coefficients in t. ii. Then, solve the corresponding homogeneous equation in part b. (i) as a homogeneous equation with constant coefficients. iii. For the equation in part b. (), determine yp by the method of undetermined coefficients. iv. Compare your results in parts a. and b. C. Hence, solve the given initial value problem. Consider the following initial value problem:x²y" + 7xy' + 9y = 27 ln x , x > 0y(1) = 1,y'(1) = -4(1)a. i. Solve the corresponding homogeneous equation as a Cauchy-Euler equation.ii. Determine y, by the method of variation of parameters.b. i. By using the following change of variables:x = etShow that Eq. (1) becomes a non-homogeneous equation with constantcoefficients in t.ii. Then, solve the corresponding homogeneous equation in part b. (i) as ahomogeneous equation with constant coefficients.iii. For the equation in part b. (i), determine y, by the method of undeterminedcoefficients.iv. Compare your results in parts a. and b.c. Hence, solve the given initial value problem.

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Answered: x²y" + 7xy' + 9y = 27 In x , x > 0 y(1)…

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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Consider the following initial value problem: (1) x2y" + 7xy' +9y = 27 In x , x > 0 y(1) = 1, y'(1) = -4 a. i. Solve the corresponding homogeneous equation as a Cauchy-Euler equation. ii. Determine yp by the method of variation of parameters. b. i. By using the following change of variables: x = et Show that Eq. (1) becomes a non-homogeneous equation with constant coefficients in t. ii. Then, solve the corresponding homogeneous equation in part b. (i) as a homogeneous equation with constant coefficients. iii. For the equation in part b. (), determine yp by the method of undetermined coefficients. iv. Compare your results in parts a. and b. C. Hence, solve the given initial value problem.

Consider the following initial value problem:
x²y" + 7xy' + 9y = 27 ln x , x > 0
y(1) = 1,y'(1) = -4
(1)
a. i. Solve the corresponding homogeneous equation as a Cauchy-Euler equation.
ii. Determine y, by the method of variation of parameters.
b. i. By using the following change of variables:
x = et
Show that Eq. (1) becomes a non-homogeneous equation with constant
coefficients in t.
ii. Then, solve the corresponding homogeneous equation in part b. (i) as a
homogeneous equation with constant coefficients.
iii. For the equation in part b. (i), determine y, by the method of undetermined
coefficients.
iv. Compare your results in parts a. and b.
c. Hence, solve the given initial value problem.

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