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]]
\[ \boxed {y^{\prime \prime }+\frac {y t^{2}}{4}-f \cos \left (t \right )=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 84
dsolve(diff(y(t),t$2)+(1/4*t^2)*y(t)=f*cos(t),y(t), singsol=all)
\[ y \left (t \right ) = \sqrt {t}\, \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) c_{2} +\sqrt {t}\, \operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) c_{1} -\frac {f \pi \sqrt {t}\, \left (\left (\int \sqrt {t}\, \operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) \cos \left (t \right )d t \right ) \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{4}\right )-\left (\int \sqrt {t}\, \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) \cos \left (t \right )d t \right ) \operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{4}\right )\right )}{4} \]
✓ Solution by Mathematica
Time used: 14.116 (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 \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt [4]{-1} t\right ) \left (\int _1^t\frac {f \cos (K[1])}{\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt [4]{-1} K[1]\right ) \left (i K[1]+\frac {(-1)^{3/4} \operatorname {ParabolicCylinderD}\left (\frac {1}{2},(-1)^{3/4} K[1]\right )}{\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1)^{3/4} K[1]\right )}\right )-\sqrt [4]{-1} \operatorname {ParabolicCylinderD}\left (\frac {1}{2},\sqrt [4]{-1} K[1]\right )}dK[1]+c_1\right )+\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1)^{3/4} t\right ) \left (\int _1^t\frac {f \cos (K[2])}{\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1)^{3/4} K[2]\right ) \left (\frac {\sqrt [4]{-1} \operatorname {ParabolicCylinderD}\left (\frac {1}{2},\sqrt [4]{-1} K[2]\right )}{\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt [4]{-1} K[2]\right )}-i K[2]\right )-(-1)^{3/4} \operatorname {ParabolicCylinderD}\left (\frac {1}{2},(-1)^{3/4} K[2]\right )}dK[2]+c_2\right ) \\ \end{align*}