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Verify that the indicated family of functions is a solution of the given differential equation. Assume...

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Basic Math

Verify that the indicated family of functions is a solution ofthe given differential equation. Assume an appropriate intervalI of definition for each solution.

d^2y/ dx^2 − 6 dy/dx + 9y = 0;    y =c1e3x +c2xe3x  When y =c1e3x + c2xe3x,

dy
dx
=
d2y
dx2
=  .


Thus, in terms of x,

d^2y/dx^2− 6 dy/dx + 9y=

+ 9(c1e3x +c2xe3x)

=  .

2) In this problem, y = c1ex +c2e−x is a two-parameter family of solutionsof the second-order DE

y'' − y = 0. Find a solution of the second-order IVP consistingof this differential equation and the given initial conditions.

y(−1) = 7,    y'(−1) = −7

y =

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