13.10.9 problem 11
Internal
problem
ID
[2410]
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
Date
solved
:
Monday, January 27, 2025 at 05:51:58 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }+\frac {t^{2} y}{4}&=f \cos \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.036 (sec). Leaf size: 81
dsolve(diff(y(t),t$2)+(1/4*t^2)*y(t)=f*cos(t),y(t), singsol=all)
\[
y = -\frac {\sqrt {t}\, \left (f \pi \left (\int \sqrt {t}\, \operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) \cos \left (t \right )d t \right ) \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{4}\right )-f \pi \left (\int \sqrt {t}\, \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) \cos \left (t \right )d t \right ) \operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{4}\right )-4 \operatorname {BesselJ}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) c_2 -4 \operatorname {BesselY}\left (\frac {1}{4}, \frac {t^{2}}{4}\right ) c_1 \right )}{4}
\]
✓ Solution by Mathematica
Time used: 42.309 (sec). Leaf size: 250
DSolve[D[y[t],{t,2}]+(1/4*t^2)*y[t]==f*Cos[t],y[t],t,IncludeSingularSolutions -> True]
\[
y(t)\to \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt [4]{-1} t\right ) \left (\int _1^t-\frac {i f \cos (K[1]) \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1)^{3/4} K[1]\right )}{(-1)^{3/4} \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1)^{3/4} K[1]\right ) \operatorname {ParabolicCylinderD}\left (\frac {1}{2},\sqrt [4]{-1} K[1]\right )+\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt [4]{-1} K[1]\right ) \left (K[1] \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1)^{3/4} K[1]\right )+\sqrt [4]{-1} \operatorname {ParabolicCylinderD}\left (\frac {1}{2},(-1)^{3/4} K[1]\right )\right )}dK[1]+c_1\right )+\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1)^{3/4} t\right ) \left (\int _1^t\frac {i f \cos (K[2]) \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt [4]{-1} K[2]\right )}{(-1)^{3/4} \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1)^{3/4} K[2]\right ) \operatorname {ParabolicCylinderD}\left (\frac {1}{2},\sqrt [4]{-1} K[2]\right )+\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt [4]{-1} K[2]\right ) \left (K[2] \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1)^{3/4} K[2]\right )+\sqrt [4]{-1} \operatorname {ParabolicCylinderD}\left (\frac {1}{2},(-1)^{3/4} K[2]\right )\right )}dK[2]+c_2\right )
\]