3. Consider the non-homogeneous Cauchy-Euler equation ax²y" + bxy' + cy = f(x) (a) Let y₁(x) and 32(2) be solutions to the associated homogeneous equation, and let y,(r) be a solution to the non-homogeneous equation. Is y(x) = C₁y₁(x) + C2y2(x) + yp(x) the general solution to the non-homogeneous ODE? Prove or provide a counterexample. (b) Solve x²y" + xy' - 9y = 5e²¹ using any method.

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,…

Find the general solution of the Euler equation on (0, ∞) Note: Use b and d for constants c₁ and C₂ y(x) x²y"+3xy-3y = 0

Find a particular solution to y" +2y'+y = 1 (B) y, = 21-1 (A) Yp (D) y,=1 (C) y,=2r-2 (E) y,=t-2 Select one: О а. А O b. B O c. C O d. E O e. D