Sr Examen
Ecuación diferencial ay'=(1-t)y-y^2
El profesor se sorprenderá mucho al ver tu solución correcta😉
⌨
Solución
Ha introducido
[src]
d 2 a*--(y(t)) = - y (t) + (1 - t)*y(t) dt
$$a \frac{d}{d t} y{\left(t \right)} = \left(1 - t\right) y{\left(t \right)} - y^{2}{\left(t \right)}$$
a*y' = (1 - t)*y - y^2
Respuesta
[src]
/ / / 2 \ \ \ / / / / 2 \ \ / 2 \\ \
| | | (1 - 2*C1) 2*C1*(1 - C1)| 6*C1*(1 - C1)*(-1 + 2*C1)| | | | | | (1 - 2*C1) 2*C1*(1 - C1)| 6*C1*(1 - C1)*(-1 + 2*C1)| | (1 - 2*C1) 6*(1 - 2*C1)*(-1 + 2*C1) 8*C1*(1 - C1)|| |
/ / 2 \\ | 2 (1 - C1)*|-2 + 6*C1 + (1 - 2*C1)*|-1 + ----------- - -------------| + -------------------------| | | / 2 \ | 2 (1 - 2*C1)*|-2 + 6*C1 + (1 - 2*C1)*|-1 + ----------- - -------------| + -------------------------| 2*C1*(1 - C1)*|4 - ----------- + ------------------------ + -------------|| / 2 \ |
2 / (1 - C1)*(1 - 2*C1)\ 3 | | (1 - 2*C1) 2*C1*(1 - C1)|| 4 | (1 - 2*C1) \ \ a a / a / 2*C1*(1 - C1) 2*(1 - C1)*(-1 + 3*C1)| 5 | | (1 - 2*C1) 6*C1*(-1 + 2*C1) 2*(1 - 2*C1)*(-1 + 3*C1) 8*C1*(1 - C1)| | (1 - 2*C1) \ \ a a / a / 6*C1*(-1 + 2*C1) 2*(1 - 2*C1)*(-1 + 3*C1) 8*C1*(1 - C1) \ a a a /| | (1 - 2*C1) 2*C1*(1 - C1)| 6*C1*(1 - C1)*(-1 + 2*C1)|
C1*t *|-1 + -------------------| C1*t *|-2 + 3*C1 + (1 - C1)*|-1 + ----------- - -------------|| C1*t *|3 - ----------- + ------------------------------------------------------------------------------------------------ + ------------- + ----------------------| C1*t *|8 - 20*C1 + (1 - C1)*|3 - ----------- - ---------------- + ------------------------ + -------------| + (1 - C1)*|3 - ----------- + -------------------------------------------------------------------------------------------------- - ---------------- + ------------------------ + ------------- + --------------------------------------------------------------------------| - (1 - 2*C1)*|-1 + ----------- - -------------| - -------------------------|
C1*t*(1 - C1) \ a / \ \ a a // \ a a a a / \ \ a a a a / \ a a a a a a / \ a a / a / / 6\
y(t) = C1 + ------------- + -------------------------------- + --------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + O\t /
a 2*a 2 2 3
6*a 24*a 120*a
$$y{\left(t \right)} = \frac{C_{1} t^{5} \left(- 20 C_{1} - \frac{6 C_{1} \left(1 - C_{1}\right) \left(2 C_{1} - 1\right)}{a} - \left(1 - 2 C_{1}\right) \left(- \frac{2 C_{1} \left(1 - C_{1}\right)}{a} - 1 + \frac{\left(1 - 2 C_{1}\right)^{2}}{a}\right) + \left(1 - C_{1}\right) \left(\frac{8 C_{1} \left(1 - C_{1}\right)}{a} - \frac{6 C_{1} \left(2 C_{1} - 1\right)}{a} + 3 - \frac{\left(1 - 2 C_{1}\right)^{2}}{a} + \frac{2 \left(1 - 2 C_{1}\right) \left(3 C_{1} - 1\right)}{a}\right) + \left(1 - C_{1}\right) \left(\frac{2 C_{1} \left(1 - C_{1}\right) \left(\frac{8 C_{1} \left(1 - C_{1}\right)}{a} + 4 - \frac{\left(1 - 2 C_{1}\right)^{2}}{a} + \frac{6 \left(1 - 2 C_{1}\right) \left(2 C_{1} - 1\right)}{a}\right)}{a} + \frac{8 C_{1} \left(1 - C_{1}\right)}{a} - \frac{6 C_{1} \left(2 C_{1} - 1\right)}{a} + 3 - \frac{\left(1 - 2 C_{1}\right)^{2}}{a} + \frac{2 \left(1 - 2 C_{1}\right) \left(3 C_{1} - 1\right)}{a} + \frac{\left(1 - 2 C_{1}\right) \left(6 C_{1} + \frac{6 C_{1} \left(1 - C_{1}\right) \left(2 C_{1} - 1\right)}{a} + \left(1 - 2 C_{1}\right) \left(- \frac{2 C_{1} \left(1 - C_{1}\right)}{a} - 1 + \frac{\left(1 - 2 C_{1}\right)^{2}}{a}\right) - 2\right)}{a}\right) + 8\right)}{120 a^{3}} + \frac{C_{1} t^{3} \left(3 C_{1} + \left(1 - C_{1}\right) \left(- \frac{2 C_{1} \left(1 - C_{1}\right)}{a} - 1 + \frac{\left(1 - 2 C_{1}\right)^{2}}{a}\right) - 2\right)}{6 a^{2}} + \frac{C_{1} t^{4} \left(\frac{2 C_{1} \left(1 - C_{1}\right)}{a} + 3 - \frac{\left(1 - 2 C_{1}\right)^{2}}{a} + \frac{2 \left(1 - C_{1}\right) \left(3 C_{1} - 1\right)}{a} + \frac{\left(1 - C_{1}\right) \left(6 C_{1} + \frac{6 C_{1} \left(1 - C_{1}\right) \left(2 C_{1} - 1\right)}{a} + \left(1 - 2 C_{1}\right) \left(- \frac{2 C_{1} \left(1 - C_{1}\right)}{a} - 1 + \frac{\left(1 - 2 C_{1}\right)^{2}}{a}\right) - 2\right)}{a}\right)}{24 a^{2}} + \frac{C_{1} t \left(1 - C_{1}\right)}{a} + \frac{C_{1} t^{2} \left(-1 + \frac{\left(1 - 2 C_{1}\right) \left(1 - C_{1}\right)}{a}\right)}{2 a} + C_{1} + O\left(t^{6}\right)$$
Clasificación
1st power series
lie group