76.11.6 problem 6

Internal problem ID [17550]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.1 (Definitions and examples). Problems at page 214
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 10:43:19 AM
CAS classification : unknown

\begin{align*} y^{\prime \prime }-t y&=\frac {1}{\pi } \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 90

dsolve(diff(y(t),t$2)-t*y(t)=1/Pi,y(t), singsol=all)
 

\[ y = \frac {4 t \pi \left (-3^{{5}/{6}} \operatorname {AiryAi}\left (t \right )+\operatorname {AiryBi}\left (t \right ) 3^{{1}/{3}}\right ) \operatorname {hypergeom}\left (\left [\frac {1}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {t^{3}}{9}\right )-3 \Gamma \left (\frac {2}{3}\right ) \left (t^{2} \Gamma \left (\frac {2}{3}\right ) \left (3^{{2}/{3}} \operatorname {AiryAi}\left (t \right )+\operatorname {AiryBi}\left (t \right ) 3^{{1}/{6}}\right ) \operatorname {hypergeom}\left (\left [\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {t^{3}}{9}\right )-4 \pi \left (\operatorname {AiryBi}\left (t \right ) c_{1} +\operatorname {AiryAi}\left (t \right ) c_{2} \right )\right )}{12 \Gamma \left (\frac {2}{3}\right ) \pi } \]

✓ Solution by Mathematica

Time used: 0.038 (sec). Leaf size: 44

DSolve[D[y[t],{t,2}]-t*y[t]==1/Pi,y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \operatorname {AiryBi}(t) \int _1^t\operatorname {AiryAi}(K[2])dK[2]+\operatorname {AiryAi}(t) \int _1^t-\operatorname {AiryBi}(K[1])dK[1]+c_1 \operatorname {AiryAi}(t)+c_2 \operatorname {AiryBi}(t) \]