Sr Examen
Ecuación diferencial y'''-2y''+ay'+by=0
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Solución
Ha introducido
[src]
2 3
d d d
- 2*---(y(x)) + a*--(y(x)) + b*y(x) + ---(y(x)) = 0
2 dx 3
dx dx
$$a \frac{d}{d x} y{\left(x \right)} + b y{\left(x \right)} - 2 \frac{d^{2}}{d x^{2}} y{\left(x \right)} + \frac{d^{3}}{d x^{3}} y{\left(x \right)} = 0$$
a*y' + b*y - 2*y'' + y''' = 0
Respuesta
[src]
/ ____________________________________________________________ \ / ____________________________________________________________ ____________________________________________________________\ / ____________________________________________________________ ____________________________________________________________\
| / _____________________________________ | | / _____________________________________ / _____________________________________ | | / _____________________________________ / _____________________________________ |
| / / 2 3 | | / / 2 3 / / 2 3 | | / / 2 3 / / 2 3 |
| / \/ (-16 + 18*a + 27*b) - 4*(4 - 3*a) 27*b | | / \/ (-16 + 18*a + 27*b) - 4*(4 - 3*a) 27*b ___ / \/ (-16 + 18*a + 27*b) - 4*(4 - 3*a) 27*b | | / \/ (-16 + 18*a + 27*b) - 4*(4 - 3*a) 27*b ___ / \/ (-16 + 18*a + 27*b) - 4*(4 - 3*a) 27*b |
| 3 / -8 + ---------------------------------------- + 9*a + ---- | | 3 / -8 + ---------------------------------------- + 9*a + ---- I*\/ 3 *3 / -8 + ---------------------------------------- + 9*a + ---- | | 3 / -8 + ---------------------------------------- + 9*a + ---- I*\/ 3 *3 / -8 + ---------------------------------------- + 9*a + ---- |
|2 4 \/ 2 2 a | |2 \/ 2 2 8 2*a \/ 2 2 | |2 \/ 2 2 8 2*a \/ 2 2 |
x*|- - -------------------------------------------------------------------- - ------------------------------------------------------------------ + ------------------------------------------------------------------| x*|- + ------------------------------------------------------------------ - ----------------------------------------------------------------------------------- + --------------------------------------------------------------------------------- + --------------------------------------------------------------------------| x*|- + ------------------------------------------------------------------ - ----------------------------------------------------------------------------------- + --------------------------------------------------------------------------------- - --------------------------------------------------------------------------|
|3 ____________________________________________________________ 3 ____________________________________________________________| |3 6 ____________________________________________________________ ____________________________________________________________ 6 | |3 6 ____________________________________________________________ ____________________________________________________________ 6 |
| / _____________________________________ / _____________________________________ | | / _____________________________________ / _____________________________________ | | / _____________________________________ / _____________________________________ |
| / / 2 3 / / 2 3 | | / / 2 3 / / 2 3 | | / / 2 3 / / 2 3 |
| / \/ (-16 + 18*a + 27*b) - 4*(4 - 3*a) 27*b / \/ (-16 + 18*a + 27*b) - 4*(4 - 3*a) 27*b | | / ___\ / \/ (-16 + 18*a + 27*b) - 4*(4 - 3*a) 27*b / ___\ / \/ (-16 + 18*a + 27*b) - 4*(4 - 3*a) 27*b | | / ___\ / \/ (-16 + 18*a + 27*b) - 4*(4 - 3*a) 27*b / ___\ / \/ (-16 + 18*a + 27*b) - 4*(4 - 3*a) 27*b |
| 3*3 / -8 + ---------------------------------------- + 9*a + ---- 3 / -8 + ---------------------------------------- + 9*a + ---- | | 3*\-1 - I*\/ 3 /*3 / -8 + ---------------------------------------- + 9*a + ---- \-1 - I*\/ 3 /*3 / -8 + ---------------------------------------- + 9*a + ---- | | 3*\-1 + I*\/ 3 /*3 / -8 + ---------------------------------------- + 9*a + ---- \-1 + I*\/ 3 /*3 / -8 + ---------------------------------------- + 9*a + ---- |
\ \/ 2 2 \/ 2 2 / \ \/ 2 2 \/ 2 2 / \ \/ 2 2 \/ 2 2 /
y(x) = C1*e + C2*e + C3*e
$$y{\left(x \right)} = C_{1} e^{x \left(\frac{a}{\sqrt[3]{9 a + \frac{27 b}{2} + \frac{\sqrt{- 4 \left(4 - 3 a\right)^{3} + \left(18 a + 27 b - 16\right)^{2}}}{2} - 8}} - \frac{\sqrt[3]{9 a + \frac{27 b}{2} + \frac{\sqrt{- 4 \left(4 - 3 a\right)^{3} + \left(18 a + 27 b - 16\right)^{2}}}{2} - 8}}{3} + \frac{2}{3} - \frac{4}{3 \sqrt[3]{9 a + \frac{27 b}{2} + \frac{\sqrt{- 4 \left(4 - 3 a\right)^{3} + \left(18 a + 27 b - 16\right)^{2}}}{2} - 8}}\right)} + C_{2} e^{x \left(\frac{2 a}{\left(-1 - \sqrt{3} i\right) \sqrt[3]{9 a + \frac{27 b}{2} + \frac{\sqrt{- 4 \left(4 - 3 a\right)^{3} + \left(18 a + 27 b - 16\right)^{2}}}{2} - 8}} + \frac{\sqrt[3]{9 a + \frac{27 b}{2} + \frac{\sqrt{- 4 \left(4 - 3 a\right)^{3} + \left(18 a + 27 b - 16\right)^{2}}}{2} - 8}}{6} + \frac{\sqrt{3} i \sqrt[3]{9 a + \frac{27 b}{2} + \frac{\sqrt{- 4 \left(4 - 3 a\right)^{3} + \left(18 a + 27 b - 16\right)^{2}}}{2} - 8}}{6} + \frac{2}{3} - \frac{8}{3 \left(-1 - \sqrt{3} i\right) \sqrt[3]{9 a + \frac{27 b}{2} + \frac{\sqrt{- 4 \left(4 - 3 a\right)^{3} + \left(18 a + 27 b - 16\right)^{2}}}{2} - 8}}\right)} + C_{3} e^{x \left(\frac{2 a}{\left(-1 + \sqrt{3} i\right) \sqrt[3]{9 a + \frac{27 b}{2} + \frac{\sqrt{- 4 \left(4 - 3 a\right)^{3} + \left(18 a + 27 b - 16\right)^{2}}}{2} - 8}} + \frac{\sqrt[3]{9 a + \frac{27 b}{2} + \frac{\sqrt{- 4 \left(4 - 3 a\right)^{3} + \left(18 a + 27 b - 16\right)^{2}}}{2} - 8}}{6} - \frac{\sqrt{3} i \sqrt[3]{9 a + \frac{27 b}{2} + \frac{\sqrt{- 4 \left(4 - 3 a\right)^{3} + \left(18 a + 27 b - 16\right)^{2}}}{2} - 8}}{6} + \frac{2}{3} - \frac{8}{3 \left(-1 + \sqrt{3} i\right) \sqrt[3]{9 a + \frac{27 b}{2} + \frac{\sqrt{- 4 \left(4 - 3 a\right)^{3} + \left(18 a + 27 b - 16\right)^{2}}}{2} - 8}}\right)}$$
Clasificación
nth linear constant coeff homogeneous