74.12.58 problem 64

Internal problem ID [16365]
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 : Tuesday, January 28, 2025 at 09:07:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} \left (\ln \left (t \right )-1\right ) y^{\prime \prime }-y^{\prime } t +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*}

✓ Solution by Maple

Time used: 0.148 (sec). Leaf size: 20

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

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

✓ Solution by Mathematica

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

DSolve[{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],{y[1]==0,Derivative[1][y][1]==0}},y[t],t,IncludeSingularSolutions -> True]
 

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