Sr Examen
Ecuación diferencial a^2*y''+a*y'-y=-2*a^2*cos(x)+a
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Solución
Ha introducido
[src]
2
d 2 d 2
-y(x) + a*--(y(x)) + a *---(y(x)) = a - 2*a *cos(x)
dx 2
dx
$$a^{2} \frac{d^{2}}{d x^{2}} y{\left(x \right)} + a \frac{d}{d x} y{\left(x \right)} - y{\left(x \right)} = - 2 a^{2} \cos{\left(x \right)} + a$$
a^2*y'' + a*y' - y = -2*a^2*cos(x) + a
Respuesta
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/ ___\ / ___\
x*\-1 + \/ 5 / x*\-1 - \/ 5 /
-------------- -------------- 3 2 4
2*a 2*a 2*a *sin(x) 2*a *cos(x) 2*a *cos(x)
y(x) = -a + C1*e + C2*e - ------------- + ------------- + -------------
4 2 4 2 4 2
1 + a + 3*a 1 + a + 3*a 1 + a + 3*a
$$y{\left(x \right)} = C_{1} e^{\frac{x \left(-1 + \sqrt{5}\right)}{2 a}} + C_{2} e^{\frac{x \left(- \sqrt{5} - 1\right)}{2 a}} + \frac{2 a^{4} \cos{\left(x \right)}}{a^{4} + 3 a^{2} + 1} - \frac{2 a^{3} \sin{\left(x \right)}}{a^{4} + 3 a^{2} + 1} + \frac{2 a^{2} \cos{\left(x \right)}}{a^{4} + 3 a^{2} + 1} - a$$
Clasificación
nth linear constant coeff undetermined coefficients
nth linear constant coeff variation of parameters
nth linear constant coeff variation of parameters Integral