Sr Examen
Ecuación diferencial p*y'+q*y+y''=e^(a*x)*(msin(b*x)+ncos(b*x))
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Solución
Ha introducido
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2
d d a*x
p*--(y(x)) + q*y(x) + ---(y(x)) = (m*sin(b*x) + n*cos(b*x))*e
dx 2
dx
$$p \frac{d}{d x} y{\left(x \right)} + q y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = \left(m \sin{\left(b x \right)} + n \cos{\left(b x \right)}\right) e^{a x}$$
p*y' + q*y + y'' = (m*sin(b*x) + n*cos(b*x))*exp(a*x)
Respuesta
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/ __________\ / __________ \
| / 2 | | / 2 |
x*\-p - \/ p - 4*q / x*\\/ p - 4*q - p/
---------------------- --------------------- a*x 2 a*x a*x 2 a*x 2 a*x 2 a*x a*x a*x a*x a*x a*x a*x
2 2 m*q*e *sin(b*x) m*a *e *sin(b*x) n*q*cos(b*x)*e n*a *cos(b*x)*e m*b *e *sin(b*x) n*b *cos(b*x)*e a*m*p*e *sin(b*x) a*n*p*cos(b*x)*e b*n*p*e *sin(b*x) b*m*p*cos(b*x)*e 2*a*b*m*cos(b*x)*e 2*a*b*n*e *sin(b*x)
y(x) = C1*e + C2*e + ------------------------------------ + ------------------------------------ + ------------------------------------ + ------------------------------------ - ------------------------------------ - ------------------------------------ + ------------------------------------ + ------------------------------------ + ------------------------------------ - ------------------------------------ - ------------------------------------ + ------------------------------------
2 2 2 2 2 2 2 2 2 2 2 2
/ 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2
\q + a - b + a*p/ + b *(p + 2*a) \q + a - b + a*p/ + b *(p + 2*a) \q + a - b + a*p/ + b *(p + 2*a) \q + a - b + a*p/ + b *(p + 2*a) \q + a - b + a*p/ + b *(p + 2*a) \q + a - b + a*p/ + b *(p + 2*a) \q + a - b + a*p/ + b *(p + 2*a) \q + a - b + a*p/ + b *(p + 2*a) \q + a - b + a*p/ + b *(p + 2*a) \q + a - b + a*p/ + b *(p + 2*a) \q + a - b + a*p/ + b *(p + 2*a) \q + a - b + a*p/ + b *(p + 2*a)
$$y{\left(x \right)} = C_{1} e^{\frac{x \left(- p - \sqrt{p^{2} - 4 q}\right)}{2}} + C_{2} e^{\frac{x \left(- p + \sqrt{p^{2} - 4 q}\right)}{2}} + \frac{a^{2} m e^{a x} \sin{\left(b x \right)}}{b^{2} \left(2 a + p\right)^{2} + \left(a^{2} + a p - b^{2} + q\right)^{2}} + \frac{a^{2} n e^{a x} \cos{\left(b x \right)}}{b^{2} \left(2 a + p\right)^{2} + \left(a^{2} + a p - b^{2} + q\right)^{2}} - \frac{2 a b m e^{a x} \cos{\left(b x \right)}}{b^{2} \left(2 a + p\right)^{2} + \left(a^{2} + a p - b^{2} + q\right)^{2}} + \frac{2 a b n e^{a x} \sin{\left(b x \right)}}{b^{2} \left(2 a + p\right)^{2} + \left(a^{2} + a p - b^{2} + q\right)^{2}} + \frac{a m p e^{a x} \sin{\left(b x \right)}}{b^{2} \left(2 a + p\right)^{2} + \left(a^{2} + a p - b^{2} + q\right)^{2}} + \frac{a n p e^{a x} \cos{\left(b x \right)}}{b^{2} \left(2 a + p\right)^{2} + \left(a^{2} + a p - b^{2} + q\right)^{2}} - \frac{b^{2} m e^{a x} \sin{\left(b x \right)}}{b^{2} \left(2 a + p\right)^{2} + \left(a^{2} + a p - b^{2} + q\right)^{2}} - \frac{b^{2} n e^{a x} \cos{\left(b x \right)}}{b^{2} \left(2 a + p\right)^{2} + \left(a^{2} + a p - b^{2} + q\right)^{2}} - \frac{b m p e^{a x} \cos{\left(b x \right)}}{b^{2} \left(2 a + p\right)^{2} + \left(a^{2} + a p - b^{2} + q\right)^{2}} + \frac{b n p e^{a x} \sin{\left(b x \right)}}{b^{2} \left(2 a + p\right)^{2} + \left(a^{2} + a p - b^{2} + q\right)^{2}} + \frac{m q e^{a x} \sin{\left(b x \right)}}{b^{2} \left(2 a + p\right)^{2} + \left(a^{2} + a p - b^{2} + q\right)^{2}} + \frac{n q e^{a x} \cos{\left(b x \right)}}{b^{2} \left(2 a + p\right)^{2} + \left(a^{2} + a p - b^{2} + q\right)^{2}}$$
Clasificación
nth linear constant coeff undetermined coefficients
nth linear constant coeff variation of parameters
nth linear constant coeff variation of parameters Integral