Find the folution to Hhe initial value problem -Please Write the folution with the computer.

Answered: Find the Bolution to Hhe initial value…

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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Find the folution to Hhe initial value problem -
Please Write the folution with the computer.

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Step 1

Solution of Linear Differential Equation with Constant Coefficients:

The general form of linear ordinary differential equation with constant coefficients is given by

$y^{(n)} + a_{1} y^{(n - 1)} + . . . + a_{n - 1} y' + a_{n} y = Q (t)$ ....(i)

with constant coefficients $a_{i}$ .

Expressed as a linear differential operator, the equation is $f (D) y = Q (t)$ , where

$f (D) = D^{n} + a_{1} D^{n - 1} + . . . + a_{n - 1} D + a_{n}$

Then the auxiliary equation is

$m^{n} + a_{1} m^{n - 1} + . . . + a_{n - 1} m + a_{n} = 0$

Solving this we get $y_{c}$ that is the general solution to the associated homogeneous equation and

then particular solution $y_{p} = \frac{1}{f (D)} Q (t)$

Thus the general solution of (i) is $y = y_{c} + y_{p}$
When we solve a linear differential equation of order n, we will have n different constants in our general solution. For this reason, we will need n initial values to find the solution to a given initial value problem.

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