Internal problem ID [819]
Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima,
Meade
Section: Chapter 4.1, Higher order linear differential equations. General theory. page 173
Problem number: 17.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {t y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }+y t=0} \end {gather*}
✓ Solution by Maple
Time used: 0.08 (sec). Leaf size: 183
dsolve(t*diff(y(t),t$3)+2*diff(y(t),t$2)-diff(y(t),t)+t*y(t)=0,y(t), singsol=all)
\[ y \relax (t ) = c_{1} \KummerM \left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right ) {\mathrm e}^{-\frac {t \left (i \sqrt {3}-1\right )}{2}}+c_{2} \KummerU \left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right ) {\mathrm e}^{-\frac {t \left (i \sqrt {3}-1\right )}{2}}+c_{3} {\mathrm e}^{-\frac {t \left (i \sqrt {3}-1\right )}{2}} \left (\KummerU \left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right ) \left (\int \KummerM \left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right ) {\mathrm e}^{-\frac {t \left (i \sqrt {3}+3\right )}{2}}d t \right )-\left (\int \KummerU \left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right ) {\mathrm e}^{-\frac {t \left (i \sqrt {3}+3\right )}{2}}d t \right ) \KummerM \left (\frac {1}{2}-\frac {i \sqrt {3}}{6}, 1, i \sqrt {3}\, t \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.639 (sec). Leaf size: 452
DSolve[t*y'''[t]+2*y''[t]-y'[t]+t*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to e^{\frac {1}{2} \left (t-i \sqrt {3} t\right )} \left (\text {HypergeometricU}\left (\frac {1}{6} \left (3-i \sqrt {3}\right ),1,i \sqrt {3} t\right ) \left (c_3 \int _1^t\frac {(-1)^{2/3} e^{\frac {1}{2} i \left (3 i+\sqrt {3}\right ) K[1]} \text {LaguerreL}\left (\frac {1}{6} i \left (3 i+\sqrt {3}\right ),i \sqrt {3} K[1]\right )}{K[1] \left (\, _1F_1\left (\frac {1}{6} \left (9-i \sqrt {3}\right );2;i \sqrt {3} K[1]\right ) \text {HypergeometricU}\left (\frac {1}{6} \left (3-i \sqrt {3}\right ),1,i \sqrt {3} K[1]\right )+\text {HypergeometricU}\left (\frac {1}{6} \left (9-i \sqrt {3}\right ),2,i \sqrt {3} K[1]\right ) \text {LaguerreL}\left (\frac {1}{6} i \left (3 i+\sqrt {3}\right ),i \sqrt {3} K[1]\right )\right )}dK[1]+c_1\right )+\text {LaguerreL}\left (\frac {1}{6} i \left (\sqrt {3}+3 i\right ),i \sqrt {3} t\right ) \left (c_3 \int _1^t-\frac {(-1)^{2/3} e^{\frac {1}{2} i \left (3 i+\sqrt {3}\right ) K[2]} \text {HypergeometricU}\left (\frac {1}{6} \left (3-i \sqrt {3}\right ),1,i \sqrt {3} K[2]\right )}{K[2] \left (\, _1F_1\left (\frac {1}{6} \left (9-i \sqrt {3}\right );2;i \sqrt {3} K[2]\right ) \text {HypergeometricU}\left (\frac {1}{6} \left (3-i \sqrt {3}\right ),1,i \sqrt {3} K[2]\right )+\text {HypergeometricU}\left (\frac {1}{6} \left (9-i \sqrt {3}\right ),2,i \sqrt {3} K[2]\right ) \text {LaguerreL}\left (\frac {1}{6} i \left (3 i+\sqrt {3}\right ),i \sqrt {3} K[2]\right )\right )}dK[2]+c_2\right )\right ) \\ \end{align*}