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Ecuación diferencial y''''=x*(1-x)-1/2*y''+y'
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Solución
Ha introducido
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2
d
---(y(x))
4 2
d dx d
---(y(x)) = - --------- + x*(1 - x) + --(y(x))
4 2 dx
dx
$$\frac{d^{4}}{d x^{4}} y{\left(x \right)} = x \left(1 - x\right) + \frac{d}{d x} y{\left(x \right)} - \frac{\frac{d^{2}}{d x^{2}} y{\left(x \right)}}{2}$$
y'''' = x*(1 - x) + y' - y''/2
Respuesta
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/ ______________ 3 ___ \ / ______________ 3 ___ \ / ______________ 3 ___ \
3 ___ |3 / _____ \/ 6 | 3 ___ | 3 / _____ \/ 6 | 3 ___ | 3 / _____ \/ 6 |
x*\/ 6 *|\/ 18 + \/ 330 - -----------------| x*\/ 6 *|- \/ 18 + \/ 330 + -----------------| / / ______________ 3 ___ \\ / / ______________ 3 ___ \\ x*\/ 6 *|- \/ 18 + \/ 330 + -----------------|
| ______________| | ______________| | 3 ___ 6 ___ | 2/3 3 / _____ 3*\/ 2 || | 3 ___ 6 ___ | 2/3 3 / _____ 3*\/ 2 || | ______________|
| 3 / _____ | | 3 / _____ | |x*\/ 2 *\/ 3 *|3 *\/ 18 + \/ 330 + -----------------|| |x*\/ 2 *\/ 3 *|3 *\/ 18 + \/ 330 + -----------------|| | 3 / _____ |
\ \/ 18 + \/ 330 / \ \/ 18 + \/ 330 / | | ______________|| | | ______________|| \ \/ 18 + \/ 330 /
3 ----------------------------------------------- ------------------------------------------------- | | 3 / _____ || | | 3 / _____ || -------------------------------------------------
x 6 12 | \ \/ 18 + \/ 330 /| | \ \/ 18 + \/ 330 /| 12
y(x) = C1 + -- + C4*e + C2*e *sin|----------------------------------------------------------| + C3*cos|----------------------------------------------------------|*e
3 \ 12 / \ 12 /
$$y{\left(x \right)} = C_{1} + C_{2} e^{\frac{\sqrt[3]{6} x \left(- \sqrt[3]{18 + \sqrt{330}} + \frac{\sqrt[3]{6}}{\sqrt[3]{18 + \sqrt{330}}}\right)}{12}} \sin{\left(\frac{\sqrt[3]{2} \sqrt[6]{3} x \left(\frac{3 \sqrt[3]{2}}{\sqrt[3]{18 + \sqrt{330}}} + 3^{\frac{2}{3}} \sqrt[3]{18 + \sqrt{330}}\right)}{12} \right)} + C_{3} e^{\frac{\sqrt[3]{6} x \left(- \sqrt[3]{18 + \sqrt{330}} + \frac{\sqrt[3]{6}}{\sqrt[3]{18 + \sqrt{330}}}\right)}{12}} \cos{\left(\frac{\sqrt[3]{2} \sqrt[6]{3} x \left(\frac{3 \sqrt[3]{2}}{\sqrt[3]{18 + \sqrt{330}}} + 3^{\frac{2}{3}} \sqrt[3]{18 + \sqrt{330}}\right)}{12} \right)} + C_{4} e^{\frac{\sqrt[3]{6} x \left(- \frac{\sqrt[3]{6}}{\sqrt[3]{18 + \sqrt{330}}} + \sqrt[3]{18 + \sqrt{330}}\right)}{6}} + \frac{x^{3}}{3}$$
Clasificación
nth linear constant coeff undetermined coefficients
nth linear constant coeff variation of parameters
nth order reducible
nth linear constant coeff variation of parameters Integral