Internal problem ID [1762]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.4, The method of variation of parameters. Page 154
Problem number: 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {t^{2} y}{4}-f \cos \relax (t )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.065 (sec). Leaf size: 84
dsolve(diff(y(t),t$2)+(1/4*t^2)*y(t)=f*cos(t),y(t), singsol=all)
\[ y \relax (t ) = \sqrt {t}\, \BesselJ \left (\frac {1}{4}, \frac {t^{2}}{4}\right ) c_{2}+\sqrt {t}\, \BesselY \left (\frac {1}{4}, \frac {t^{2}}{4}\right ) c_{1}+\frac {\sqrt {t}\, f \pi \left (\BesselY \left (\frac {1}{4}, \frac {t^{2}}{4}\right ) \left (\int \sqrt {t}\, \BesselJ \left (\frac {1}{4}, \frac {t^{2}}{4}\right ) \cos \relax (t )d t \right )-\BesselJ \left (\frac {1}{4}, \frac {t^{2}}{4}\right ) \left (\int \sqrt {t}\, \BesselY \left (\frac {1}{4}, \frac {t^{2}}{4}\right ) \cos \relax (t )d t \right )\right )}{4} \]
✓ Solution by Mathematica
Time used: 14.834 (sec). Leaf size: 208
DSolve[y''[t]+(1/4*t^2)*y[t]==f*Cos[t],y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to D_{-\frac {1}{2}}\left (\sqrt [4]{-1} t\right ) \left (\int _1^t\frac {f \cos (K[1])}{D_{-\frac {1}{2}}\left (\sqrt [4]{-1} K[1]\right ) \left (i K[1]+\frac {(-1)^{3/4} D_{\frac {1}{2}}\left ((-1)^{3/4} K[1]\right )}{D_{-\frac {1}{2}}\left ((-1)^{3/4} K[1]\right )}\right )-\sqrt [4]{-1} D_{\frac {1}{2}}\left (\sqrt [4]{-1} K[1]\right )}dK[1]+c_1\right )+D_{-\frac {1}{2}}\left ((-1)^{3/4} t\right ) \left (\int _1^t\frac {f \cos (K[2])}{D_{-\frac {1}{2}}\left ((-1)^{3/4} K[2]\right ) \left (\frac {\sqrt [4]{-1} D_{\frac {1}{2}}\left (\sqrt [4]{-1} K[2]\right )}{D_{-\frac {1}{2}}\left (\sqrt [4]{-1} K[2]\right )}-i K[2]\right )-(-1)^{3/4} D_{\frac {1}{2}}\left ((-1)^{3/4} K[2]\right )}dK[2]+c_2\right ) \\ \end{align*}