79.2.3 problem 3

Internal problem ID [18516]
Book : Elementary Differential Equations. By R.L.E. Schwarzenberger. Chapman and Hall. London. First Edition (1969)
Section : Chapter 4. Autonomous systems. Exercises at page 69
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 11:51:40 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} t^{2} x^{\prime \prime }-2 x^{\prime } t +2 x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 11

dsolve(t^2*diff(x(t),t$2)-2*t*diff(x(t),t)+2*x(t)=0,x(t), singsol=all)
 

\[ x = t \left (c_{1} t +c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.111 (sec). Leaf size: 133

DSolve[t^2*D[x[t],{t,2}]-2*D[x[t],t]+2*x[t]==0,x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to 2^{-\frac {1}{2} i \left (\sqrt {7}-i\right )} t^{\frac {1}{2}-\frac {i \sqrt {7}}{2}} \left (c_2 t^{i \sqrt {7}} \operatorname {Hypergeometric1F1}\left (-\frac {1}{2}-\frac {i \sqrt {7}}{2},1-i \sqrt {7},-\frac {2}{t}\right )+2^{i \sqrt {7}} c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{2} i \left (i+\sqrt {7}\right ),1+i \sqrt {7},-\frac {2}{t}\right )\right ) \]