57.3.8 problem 7

Internal problem ID [14324]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.2 Antiderivatives. Exercises page 19
Problem number : 7
Date solved : Thursday, October 02, 2025 at 09:31:31 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} t x^{\prime \prime }+x^{\prime }&=1 \end{align*}

With initial conditions

\begin{align*} x \left (1\right )&=0 \\ x^{\prime }\left (1\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 9

ode:=diff(x(t),t)+t*diff(diff(x(t),t),t) = 1; 
ic:=[x(1) = 0, D(x)(1) = 2]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \ln \left (t \right )+t -1 \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 10

ode=D[t*D[x[t],t],t]==1; 
ic={x[1]==0,Derivative[1][x][1 ]==2}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to t+\log (t)-1 \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 8

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(t*Derivative(x(t), (t, 2)) + Derivative(x(t), t) - 1,0) 
ics = {x(1): 0, Subs(Derivative(x(t), t), t, 1): 2} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = t + \log {\left (t \right )} - 1 \]