15.18.11 problem 11

Internal problem ID [3254]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 35, page 157
Problem number : 11
Date solved : Sunday, March 30, 2025 at 01:25:13 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x^{\prime \prime }+t x^{\prime }&=t^{3} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 30

ode:=diff(diff(x(t),t),t)+t*diff(x(t),t) = t^3; 
dsolve(ode,x(t), singsol=all);
 

\[ x = \frac {t^{3}}{3}+\frac {c_1 \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, t}{2}\right )}{2}-2 t +c_2 \]

✓ Mathematica. Time used: 0.128 (sec). Leaf size: 38

ode=D[x[t],{t,2}]+t*D[x[t],t]==t^3; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \sqrt {\frac {\pi }{2}} c_1 \text {erf}\left (\frac {t}{\sqrt {2}}\right )+\frac {t^3}{3}-2 t+c_2 \]

✓ Sympy. Time used: 9.202 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-t**3 + t*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = C_{1} + C_{2} \operatorname {erf}{\left (\frac {\sqrt {2} t}{2} \right )} + \frac {t^{3}}{3} - 2 t \]