13.7.10 problem 10

Internal problem ID [2373]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2, linear equations with constant coefficients. Page 138
Problem number : 10
Date solved : Monday, January 27, 2025 at 05:50:34 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} t^{2} y^{\prime \prime }+\alpha t y^{\prime }+\beta y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 52

dsolve(t^2*diff(y(t),t$2)+alpha*t*diff(y(t),t)+beta*y(t)=0,y(t), singsol=all)
 

\[ y = t^{-\frac {\alpha }{2}} \sqrt {t}\, \left (t^{\frac {\sqrt {\alpha ^{2}-2 \alpha -4 \beta +1}}{2}} c_1 +t^{-\frac {\sqrt {\alpha ^{2}-2 \alpha -4 \beta +1}}{2}} c_2 \right ) \]

✓ Solution by Mathematica

Time used: 0.049 (sec). Leaf size: 57

DSolve[t^2*D[y[t],{t,2}]+\[Alpha]*t*D[y[t],t]+\[Beta]*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to t^{\frac {1}{2} \left (-\sqrt {\alpha ^2-2 \alpha -4 \beta +1}-\alpha +1\right )} \left (c_2 t^{\sqrt {\alpha ^2-2 \alpha -4 \beta +1}}+c_1\right ) \]