Initial conditions to find system of equations, Mathematics

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Solve the subsequent IVP.

y′′ + 11y′ + 24 y = 0

y (0) =0

 y′ (0)=-7

 Solution

The characteristic equation is as

r2 +11r + 24 = 0

( r + 8) ( r + 3) = 0

 Its roots are r1 = - 8 and r2 = -3 and then the general solution and its derivative is.

y (t ) = ce-8t  + c e-3t

 y′ (t ) = -8c1e -8t - 3c2e -3t

 Here, plug in the initial conditions to find the subsequent system of equations.

0 = y (0) = c1 + c2

-7 = y′ (0) = -8c1 - 3c2

Solving this system provides c1 = 7and c2 = - 7. The actual solution to the differential equation is so y (t ) = (7/5) e-8t  - (7/5) e-3t


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