Sr Examen
Ecuación diferencial by'''-y''-y'+y=0
El profesor se sorprenderá mucho al ver tu solución correcta😉
⌨
Solución
Ha introducido
[src]
2 3
d d d
- --(y(x)) - ---(y(x)) + b*---(y(x)) + y(x) = 0
dx 2 3
dx dx
$$b \frac{d^{3}}{d x^{3}} y{\left(x \right)} + y{\left(x \right)} - \frac{d}{d x} y{\left(x \right)} - \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0$$
b*y''' + y - y' - y'' = 0
Respuesta
[src]
/ ____________________________________________________________ \ / ____________________________________________________________ ____________________________________________________________\ / ____________________________________________________________ ____________________________________________________________\
| / _____________________________ | | / _____________________________ / _____________________________ | | / _____________________________ / _____________________________ |
| / / 3 | | / / 3 / / 3 | | / / 3 / / 3 |
| / / / 1\ | | / / / 1\ / / / 1\ | | / / / 1\ / / / 1\ |
| / / 2 4*|3 + -| | | / / 2 4*|3 + -| / / 2 4*|3 + -| | | / / 2 4*|3 + -| / / 2 4*|3 + -| |
| / / / 9 2 \ \ b/ | | / / / 9 2 \ \ b/ / / / 9 2 \ \ b/ | | / / / 9 2 \ \ b/ / / / 9 2 \ \ b/ |
| / / |27 - - - --| - ---------- | | / / |27 - - - --| - ---------- / / |27 - - - --| - ---------- | | / / |27 - - - --| - ---------- / / |27 - - - --| - ---------- |
| / / | b 2| b | | / / | b 2| b / / | b 2| b | | / / | b 2| b / / | b 2| b |
| / / \ b / | | / / \ b / / / \ b / | | / / \ b / / / \ b / |
| / / --------------------------- | | / / --------------------------- / / --------------------------- | | / / --------------------------- / / --------------------------- |
| / / 2 | | / / 2 / / 2 | | / / 2 / / 2 |
| / \/ b 1 9 27 | | / \/ b 1 9 27 ___ / \/ b 1 9 27 | | / \/ b 1 9 27 ___ / \/ b 1 9 27 |
| / ---------------------------------------- - -- - ---- + --- | | / ---------------------------------------- - -- - ---- + --- I*\/ 3 * / ---------------------------------------- - -- - ---- + --- | | / ---------------------------------------- - -- - ---- + --- I*\/ 3 * / ---------------------------------------- - -- - ---- + --- |
| 3 / 2 3 2 2*b | | 3 / 2 3 2 2*b 3 / 2 3 2 2*b | | 3 / 2 3 2 2*b 3 / 2 3 2 2*b |
| \/ b 2*b 1 1 1 | | 1 \/ b 2*b 2 2 \/ b 2*b | | 1 \/ b 2*b 2 2 \/ b 2*b |
x*|- --------------------------------------------------------------------------- + --- - ----------------------------------------------------------------------------- - --------------------------------------------------------------------------------| x*|--- + --------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------| x*|--- + --------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------|
| 3 3*b ____________________________________________________________ ____________________________________________________________| |3*b 6 ____________________________________________________________ ____________________________________________________________ 6 | |3*b 6 ____________________________________________________________ ____________________________________________________________ 6 |
| / _____________________________ / _____________________________ | | / _____________________________ / _____________________________ | | / _____________________________ / _____________________________ |
| / / 3 / / 3 | | / / 3 / / 3 | | / / 3 / / 3 |
| / / / 1\ / / / 1\ | | / / / 1\ / / / 1\ | | / / / 1\ / / / 1\ |
| / / 2 4*|3 + -| / / 2 4*|3 + -| | | / / 2 4*|3 + -| / / 2 4*|3 + -| | | / / 2 4*|3 + -| / / 2 4*|3 + -| |
| / / / 9 2 \ \ b/ / / / 9 2 \ \ b/ | | / / / 9 2 \ \ b/ / / / 9 2 \ \ b/ | | / / / 9 2 \ \ b/ / / / 9 2 \ \ b/ |
| / / |27 - - - --| - ---------- / / |27 - - - --| - ---------- | | / / |27 - - - --| - ---------- / / |27 - - - --| - ---------- | | / / |27 - - - --| - ---------- / / |27 - - - --| - ---------- |
| / / | b 2| b / / | b 2| b | | / / | b 2| b / / | b 2| b | | / / | b 2| b / / | b 2| b |
| / / \ b / / / \ b / | | / / \ b / / / \ b / | | / / \ b / / / \ b / |
| / / --------------------------- / / --------------------------- | | / / --------------------------- / / --------------------------- | | / / --------------------------- / / --------------------------- |
| / / 2 / / 2 | | / / 2 / / 2 | | / / 2 / / 2 |
| / \/ b 1 9 27 2 / \/ b 1 9 27 | | / ___\ / \/ b 1 9 27 2 / ___\ / \/ b 1 9 27 | | / ___\ / \/ b 1 9 27 2 / ___\ / \/ b 1 9 27 |
| b* / ---------------------------------------- - -- - ---- + --- 3*b * / ---------------------------------------- - -- - ---- + --- | | b*\-1 + I*\/ 3 /* / ---------------------------------------- - -- - ---- + --- 3*b *\-1 + I*\/ 3 /* / ---------------------------------------- - -- - ---- + --- | | b*\-1 - I*\/ 3 /* / ---------------------------------------- - -- - ---- + --- 3*b *\-1 - I*\/ 3 /* / ---------------------------------------- - -- - ---- + --- |
| 3 / 2 3 2 2*b 3 / 2 3 2 2*b | | 3 / 2 3 2 2*b 3 / 2 3 2 2*b | | 3 / 2 3 2 2*b 3 / 2 3 2 2*b |
\ \/ b 2*b \/ b 2*b / \ \/ b 2*b \/ b 2*b / \ \/ b 2*b \/ b 2*b /
y(x) = C1*e + C2*e + C3*e
$$y{\left(x \right)} = C_{1} e^{x \left(- \frac{\sqrt[3]{\frac{\sqrt{\frac{\left(27 - \frac{9}{b} - \frac{2}{b^{2}}\right)^{2} - \frac{4 \left(3 + \frac{1}{b}\right)^{3}}{b}}{b^{2}}}}{2} + \frac{27}{2 b} - \frac{9}{2 b^{2}} - \frac{1}{b^{3}}}}{3} + \frac{1}{3 b} - \frac{1}{b \sqrt[3]{\frac{\sqrt{\frac{\left(27 - \frac{9}{b} - \frac{2}{b^{2}}\right)^{2} - \frac{4 \left(3 + \frac{1}{b}\right)^{3}}{b}}{b^{2}}}}{2} + \frac{27}{2 b} - \frac{9}{2 b^{2}} - \frac{1}{b^{3}}}} - \frac{1}{3 b^{2} \sqrt[3]{\frac{\sqrt{\frac{\left(27 - \frac{9}{b} - \frac{2}{b^{2}}\right)^{2} - \frac{4 \left(3 + \frac{1}{b}\right)^{3}}{b}}{b^{2}}}}{2} + \frac{27}{2 b} - \frac{9}{2 b^{2}} - \frac{1}{b^{3}}}}\right)} + C_{2} e^{x \left(\frac{\sqrt[3]{\frac{\sqrt{\frac{\left(27 - \frac{9}{b} - \frac{2}{b^{2}}\right)^{2} - \frac{4 \left(3 + \frac{1}{b}\right)^{3}}{b}}{b^{2}}}}{2} + \frac{27}{2 b} - \frac{9}{2 b^{2}} - \frac{1}{b^{3}}}}{6} - \frac{\sqrt{3} i \sqrt[3]{\frac{\sqrt{\frac{\left(27 - \frac{9}{b} - \frac{2}{b^{2}}\right)^{2} - \frac{4 \left(3 + \frac{1}{b}\right)^{3}}{b}}{b^{2}}}}{2} + \frac{27}{2 b} - \frac{9}{2 b^{2}} - \frac{1}{b^{3}}}}{6} + \frac{1}{3 b} - \frac{2}{b \left(-1 + \sqrt{3} i\right) \sqrt[3]{\frac{\sqrt{\frac{\left(27 - \frac{9}{b} - \frac{2}{b^{2}}\right)^{2} - \frac{4 \left(3 + \frac{1}{b}\right)^{3}}{b}}{b^{2}}}}{2} + \frac{27}{2 b} - \frac{9}{2 b^{2}} - \frac{1}{b^{3}}}} - \frac{2}{3 b^{2} \left(-1 + \sqrt{3} i\right) \sqrt[3]{\frac{\sqrt{\frac{\left(27 - \frac{9}{b} - \frac{2}{b^{2}}\right)^{2} - \frac{4 \left(3 + \frac{1}{b}\right)^{3}}{b}}{b^{2}}}}{2} + \frac{27}{2 b} - \frac{9}{2 b^{2}} - \frac{1}{b^{3}}}}\right)} + C_{3} e^{x \left(\frac{\sqrt[3]{\frac{\sqrt{\frac{\left(27 - \frac{9}{b} - \frac{2}{b^{2}}\right)^{2} - \frac{4 \left(3 + \frac{1}{b}\right)^{3}}{b}}{b^{2}}}}{2} + \frac{27}{2 b} - \frac{9}{2 b^{2}} - \frac{1}{b^{3}}}}{6} + \frac{\sqrt{3} i \sqrt[3]{\frac{\sqrt{\frac{\left(27 - \frac{9}{b} - \frac{2}{b^{2}}\right)^{2} - \frac{4 \left(3 + \frac{1}{b}\right)^{3}}{b}}{b^{2}}}}{2} + \frac{27}{2 b} - \frac{9}{2 b^{2}} - \frac{1}{b^{3}}}}{6} + \frac{1}{3 b} - \frac{2}{b \left(-1 - \sqrt{3} i\right) \sqrt[3]{\frac{\sqrt{\frac{\left(27 - \frac{9}{b} - \frac{2}{b^{2}}\right)^{2} - \frac{4 \left(3 + \frac{1}{b}\right)^{3}}{b}}{b^{2}}}}{2} + \frac{27}{2 b} - \frac{9}{2 b^{2}} - \frac{1}{b^{3}}}} - \frac{2}{3 b^{2} \left(-1 - \sqrt{3} i\right) \sqrt[3]{\frac{\sqrt{\frac{\left(27 - \frac{9}{b} - \frac{2}{b^{2}}\right)^{2} - \frac{4 \left(3 + \frac{1}{b}\right)^{3}}{b}}{b^{2}}}}{2} + \frac{27}{2 b} - \frac{9}{2 b^{2}} - \frac{1}{b^{3}}}}\right)}$$
Clasificación
nth linear constant coeff homogeneous