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Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 172
Fundamentals Of Differential Equations And Boundary Value Problems

Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

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Short Answer

Solve the given initial value problem. y''-2y'+y=0;y(0)=1,y'(0)=-2

The solution of the given initial value y''-2y'+y=0 is y(t)=et-3tet when y(0)=1 and y'(0)=-2 .

See the step by step solution

Step by Step Solution

Step 1: Differentiate the value of y.

Given differential equation is y''-2y'+y=0

Let y=ert

Therefore,

y'(t)=rerty''(t)=r2ert

Step 2: Finding the general solution.

Then the auxiliary equation is r2-2r+1=0

(r-1)2=0r-1=0r=1

Therefore, the general solution is y(t)=c1et+c2tet .

Step 3: Finding the values of c1 and c2

Given initial conditions are y(0)=1 and y'(0)=-2

y(0)=c1e0+c2×0×e0 c1=1

And y'(t)=c1et+c2et+c2tet

Then,

y'(0)=c1e0+c2e0+c.0×e0c1+c2=-2

Substitute c1 in the above equation

1+c2=-2       c2=-3

Therefore, the solution is y(t)=et-3tet .

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