1.(a) Verify that y₁ = e²x and y2 = e5x are solutions ofy" - 7y' +10y = 0(A)on (-∞, ∞).(b) Verify that if c₁ and c₂ are arbitrary constants then y = = C₁e²x + c₂e5x is a solution of (A)on (-∞, ∞).

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Answered: 1. (a) Verify that y1 = : e²x and y2 =…

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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y" +p (t)y' +q(t)y = 0 (i), can be put in more suitable form for finding a solution by making a change of the independent and/or dependent variables. Determine conditions on p and q that enable this equation to be transformed into an equation with constant coefficients by a change of the independent variable. Let r = u (t) be the new independent variable. It is easy to show (島) ( + ( +p(t) + a()y = 0 (1i). dy that dt dz dy d'y and 2 d'y d²z dy dt dz The differential equation becomes dt dr dt d'y d'z de dz dy d'y , 9 and y must be proportional. If g (t) > 0, then choose the constant of proportionality to be 1. Hence, x = u (t) = [g (t)]i dt. With a chosen this way, the coefficient of in equation (ii) is also a constant, provided that the function H = g'(t) +2p(t)q(t) is a dy In order for equation (ii) to have constant coefficients, the coefficients of constant. Thus, equation (i) can be transformed into an equation with constant coefficients by a change of the independent variable,…
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1. (a) Verify that y1 = : e²x and y2 = e5x are solutions of y" - 7y' +10y = 0 (A) on (-∞, ∞). (b) Verify that if c₁ and c₂ are arbitrary constants then y = C₁e²x + c₂e5x is a solution of (A) on (-∞, ∞).