In this project we consider the special linear homogeneous differential equations called Cauchy-Euler equations of the form aot + a₁th-1 dt + a₂t"-2. dt-2 +an-it. +any=0, t>0 dt where ao,..., an are real constants. (Note that in each term the power of the monomial of t and the order of the derivative of y are identical.) A Cauchy-Euler equation can be transformed to a constant coefficient linear equation. So we can find the solutions of such an equation using the methods we know. The procedure is described as follows. (i) Set t = t(x) = e* with z being a new independent variable on the interval (-∞0,00). Let u(x) = y(t(x)) = y(e*). By applying the chain rule u(x) = (y(t)) and the fact=e=t, we have the relations dy du dt dr 2²y_d'u = du dt2 dr² dr dt³ dr³ (ii) Substituting these relations into the Cauchy-Euler equation we have a constant coeffi- cient linear differential equation for u(z). (Recall u(x)= y(t).) (iii) We are able to find a fundamental set of solutions for the resulting constant coefficient linear differential equation for u(x), say {u, (r), u₂(x),..., un(x)}. d'u du dr² (iv) Finally, using z = lnt, a fundamental set of solutions of the original Cauchy-Euler equation is {y(t), 32(t),...,n(t)}, where y(t) = u₁(Int), y(t) = u₂(Int),... Yn (t) = u, (Int), and the general solution is y=c₁y₁(t) + ₂y₂(t)+...+ CnYn (t). dt² dt Use the above procedure and the provided derivative formulas in (i) (which you do not need to derive them again) to find the general solution of the following Cauchy-Euler equations: dy +7t + y = 0, t>0