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In this project we consider the special linear homogeneous differential equations called Cauchy-Euler equations of the form aot + a₁t"-1 dt + a₂t"-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(r) = (y(t)) and the fact=e=t, we have the relations dy du dt dr d-2y dt-2 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(r). (Recall u(x) = y(t).) dt² (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),..., u(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), yn(t)}, where y(t) = u₁(Int), 32(t) = u₂(Int),... Yn (t) = u, (Int), and the general solution is y=c₁y₁ (t) + ₂y2(t)+...+ CnYn (t). 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, dt t>0

In this project we consider the special linear homogeneous differential equations called Cauchy-Euler equations of the form aot + a₁t"-1 dt + a₂t"-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(r) = (y(t)) and the fact=e=t, we have the relations dy du dt dr d-2y dt-2 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(r). (Recall u(x) = y(t).) dt² (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),..., u(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), yn(t)}, where y(t) = u₁(Int), 32(t) = u₂(Int),... Yn (t) = u, (Int), and the general solution is y=c₁y₁ (t) + ₂y2(t)+...+ CnYn (t). 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, dt t>0

Advanced Engineering Mathematics
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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In this project we consider the special linear homogeneous differential equations called Cauchy-Euler equations of the form d-ly aot + a₁th-1 +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 a 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 2²y_du dt dx' 1. 32 y dt² dt² + a₂t"-2, (ii) Substituting these relations into the Cauchy-Euler equation we have a constant coeffi- cient linear differential equation for u(r). (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 {u₁(x), u₂(x),..., un(x)}. dy +7t+y=0, dt d-2y dt-2 (iv) Finally, using z = lnt, a fundamental set of solutions of the original Cauchy-Euler equation is {y(t), 32(t), yn(t)}, where y(t) = u₁(Int), y2(t) = u₂(Int),... Yn (t) = u, (Int), and the general solution is y=c₁y₁ (t) + ₂y2(t)+...+ CnYn (t). 3t 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 du dr² dr' dt³ + 5y = 0, +52²ydy dt2 d'u dr³ dr² d'u du t>0 t>0 -8y=0, t>0 dt3 dt Hint: you may need the factorization r³+2r²-4r-8= (r+ 2)²(r-2).
Transcribed Image Text:In this project we consider the special linear homogeneous differential equations called Cauchy-Euler equations of the form d-ly aot + a₁th-1 +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 a 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 2²y_du dt dx' 1. 32 y dt² dt² + a₂t"-2, (ii) Substituting these relations into the Cauchy-Euler equation we have a constant coeffi- cient linear differential equation for u(r). (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 {u₁(x), u₂(x),..., un(x)}. dy +7t+y=0, dt d-2y dt-2 (iv) Finally, using z = lnt, a fundamental set of solutions of the original Cauchy-Euler equation is {y(t), 32(t), yn(t)}, where y(t) = u₁(Int), y2(t) = u₂(Int),... Yn (t) = u, (Int), and the general solution is y=c₁y₁ (t) + ₂y2(t)+...+ CnYn (t). 3t 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 du dr² dr' dt³ + 5y = 0, +52²ydy dt2 d'u dr³ dr² d'u du t>0 t>0 -8y=0, t>0 dt3 dt Hint: you may need the factorization r³+2r²-4r-8= (r+ 2)²(r-2).
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