2. Consider the 2nd order linear homogeneous DEx*y" – x (x+2) y' + (x+2)y = 0 for x >0.(a) Show that y=x is a solution.(b) Seek a second solution of the form y = x v(x) and find it.(c)Use the method of "variation of parameters" to obtain theGS of x’y" – x(x+2) y' + (x+2) y = x² sin x for x >0.

Answered: Image /qna-images/answer/94bd9fa0-a50f-494f-a848-99a0cfc2bcf6.jpg

Answered: 2. Consider the 2nd order linear…

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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Locate the real value solution of
3)
Find the general solution of x(x + 2) ax dy + (x + 2)y(x) = x³, dx2 given that y, (x) = x and y2 (x) = xe* are linearly independent solutions of the corresponding homogeneous equation. Use variation of parameters.
Determine whether the following equations are linear, separable, Bernoulli, or nonlinear homogenous. Explain. a) xyy'+x²+y²=0 b) xy'=y+x²y² c) y'= (2xy)/(1+3y) d) yy'=8x²y-2y²
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,…
2. (a) Use the method of undetermined coefficient to find the general solution of the non- homogeneous DE у" — 2у' + у %3 е*. - (b) Use method of variation of parameter to solve the non-homogeneous Cauchy- Euler form of DE In x ху" — у'
2. Suppose that y₁ is a solution for of the homogeneous SOLDE " + p(t)y' + q(t)y = 0. (a) Explain how to find a non constant function tv(t) such that y2(t) = v(t)y₁(t) is also a solution. (b) Explain why y₁ and y₂ are linearly independent.
A. y" + 2y' + y = 0 B. y" + 2y' + 2y = 0 C. y" + 4y = 0 D. y" – 3y' + 2y = 0 Table 1: A-D are homogeneous second order linear equations.
A. y" + 2y' + y = 0 B. y" + 2y' + 2y = 0 C. y" + 4y = 0 D. y" – 3y' + 2y = 0 Table 1: A-D are homogeneous second order linear equations.
Given an IVP d"y + an-1(x) dn-ly dy +...+ a1 (x) dx + ao (x)y = g(x) drr-1 у (хо) — Уо, у (tо) — Ул, -.., у" "(2о) — Ул-1 If the coefficients an (x), ..., ao (x) and the right hand side of the equation g(x) are continuous on an interval I and if an (x) +0 on I then the IVP has a unique solution for the point xo e I that exists on the whole interval I. Consider the IVP on the whole real line d'y - 36) dx4 dy + y= sin(x) 1 (x² dz3 x2 + 36 dx у(3) — 916, у (3) — 14, у"(3) — 8, у" (3) — 8, The Fundamental Existence Theorem for Linear Differential Equations guarantees the existence of a unique solution on the interval
True or false

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