Pregunta
Find the general solution of the given second-order differential equation.
Ask by Salinas Dawson. in the United States
Jan 24,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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The general solution is:
Solución
To find the general solution of the second-order differential equation:
we start by finding the characteristic equation associated with the differential equation. Assume a solution of the form
. Plugging this into the differential equation gives:
Dividing through by
(which is never zero):
This is a quadratic equation in
. To solve it, we calculate the discriminant:
Since the discriminant is negative (
), the roots are complex:
Letting
and
, the general solution for the differential equation with complex roots is:
Substituting the values of
and
:
where
and
are arbitrary constants determined by initial conditions.
Final Answer:
After simplifying, a general solution is e^(5⁄4 x) times (C₁ cos(√23⁄4 x) plus C₂ sin(√23⁄4 x)). Thus,
y(x) = e^(5/4x)[C₁ cos(√23/4 x) + C₂ sin(√23/4 x)]
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Como
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Mind Expander
To solve the given second-order differential equation, we start by writing the characteristic equation associated with it. The equation is:
Using the quadratic formula,
:
Here,
,
, and
:
Since the discriminant is negative, we find complex roots:
Now we can express the general solution:
Therefore, the general solution is:

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