Find a general solution to the differential equation.y'' -6y' +9y=t6e³tThe general solution is y(t) =

Answered: Find a general solution to the…

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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Question

Find a general solution to the differential equation.
y'' -6y' +9y=t6e³t
The general solution is y(t) =

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Step 1

Sol:-

$y'' - 6 y' + 9 y = t^{- 6} e^{3 t} A second order linear, non - homogeneous ODE has the form of a y'' + b y' + c y = g (x) Rewrite in the form of a Second order linear non - homogeneous ODE y'' - 6 y' + 9 y = \frac{e^{3 t}}{t^{6}} The general solution to a (x) y'' + b (x) y' + c (x) y = g (x) can be written as y = y_{h} + y_{p} y_{h} is the solution to the homogeneous ODE a (x) y'' + b (x) y' + c (x) y = 0 y_{p}, the particular solution, is any function that satisfies the non - homogeneous equation Find y_{h} by solving y'' - 6 y' + 9 y = 0 y'' - 6 y' + 9 y = 0 A second order linear, homogeneous ODE has the form of ay'' + by' + cy = 0 For an equation ay'' + by' + cy = 0, assume a solution of the form e^{γt} Rewrite the equation with y = e^{γt} ((e^{γt}))'' - 6 ((e^{γt}))' + 9 e^{γt} = 0 Simplify ((e^{γt}))'' - 6 ((e^{γt}))' + 9 e^{γt} = 0 : e^{γt} (γ^{2} - 6 γ + 9) = 0 Solve e^{γt} (γ^{2} - 6 γ + 9) = 0 : γ = 3 with multiplicity of 2 For real root γ with multiplicity 2, the general solution takes the form : y = c_{1} e^{γ t} + c_{2} {te}^{γ t} y_{h} = c_{1} e^{3 t} + c_{2} {te}^{3 t}$