74.12.58 problem 64

Internal problem ID [16294]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 64
Date solved : Monday, March 31, 2025 at 02:49:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} \left (\ln \left (t \right )-1\right ) y^{\prime \prime }-t y^{\prime }+y&=-\frac {3 \left (1+\ln \left (t \right )\right )}{4 \sqrt {t}} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0\\ y^{\prime }\left (1\right )&=0 \end{align*}

✓ Maple. Time used: 0.048 (sec). Leaf size: 20

ode:=t^2*(ln(t)-1)*diff(diff(y(t),t),t)-t*diff(y(t),t)+y(t) = -3/4*(ln(t)+1)/t^(1/2); 
ic:=y(1) = 0, D(y)(1) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \left (-\frac {1}{t^{{3}/{2}}}-\frac {3 \ln \left (t \right )}{2 t}+1\right ) t \]

✓ Mathematica. Time used: 2.306 (sec). Leaf size: 20

ode=t^2*(Log[t]-1)*D[y[t],{t,2}]-t*D[y[t],t]+y[t]==-3/4*(1+Log[t])*1/Sqrt[t]; 
ic={y[1]==0,Derivative[1][y][1]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to t-\frac {1}{\sqrt {t}}-\frac {3 \log (t)}{2} \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*(log(t) - 1)*Derivative(y(t), (t, 2)) - t*Derivative(y(t), t) + y(t) + (3*log(t)/4 + 3/4)/sqrt(t),0) 
ics = {y(1): 0, Subs(Derivative(y(t), t), t, 1): 0} 
dsolve(ode,func=y(t),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(t), t) - (t**(7/2)*(log(t) - 1)*Derivative(y(t), (t, 2)) + t**(3/2)*y(t) + 3*t*(log(t) + 1)/4)/t**(5/2) cannot be solved by the factorable group method