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]]
\[ \boxed {y^{\prime \prime }+\frac {y t^{2}}{4}=f \cos \left (t \right )} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 81
dsolve(diff(y(t),t$2)+(1/4*t^2)*y(t)=f*cos(t),y(t), singsol=all)
\[ y \left (t \right ) = \frac {\sqrt {t}\, \left (f \pi \left (\int \sqrt {t}\, \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) \cos \left (t \right )d t \right ) \operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{4}\right )-f \pi \left (\int \sqrt {t}\, \operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) \cos \left (t \right )d t \right ) \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{4}\right )+4 \operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) c_{1} +4 \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) c_{2} \right )}{4} \]
✓ Solution by Mathematica
Time used: 29.274 (sec). Leaf size: 250
DSolve[y''[t]+(1/4*t^2)*y[t]==f*Cos[t],y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt [4]{-1} t\right ) \left (\int _1^t-\frac {i f \cos (K[1]) \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1)^{3/4} K[1]\right )}{(-1)^{3/4} \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1)^{3/4} K[1]\right ) \operatorname {ParabolicCylinderD}\left (\frac {1}{2},\sqrt [4]{-1} K[1]\right )+\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt [4]{-1} K[1]\right ) \left (K[1] \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1)^{3/4} K[1]\right )+\sqrt [4]{-1} \operatorname {ParabolicCylinderD}\left (\frac {1}{2},(-1)^{3/4} K[1]\right )\right )}dK[1]+c_1\right )+\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1)^{3/4} t\right ) \left (\int _1^t\frac {i f \cos (K[2]) \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt [4]{-1} K[2]\right )}{(-1)^{3/4} \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1)^{3/4} K[2]\right ) \operatorname {ParabolicCylinderD}\left (\frac {1}{2},\sqrt [4]{-1} K[2]\right )+\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt [4]{-1} K[2]\right ) \left (K[2] \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1)^{3/4} K[2]\right )+\sqrt [4]{-1} \operatorname {ParabolicCylinderD}\left (\frac {1}{2},(-1)^{3/4} K[2]\right )\right )}dK[2]+c_2\right ) \]