80.3.14 problem 14

Internal problem ID [21178]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 3. First order nonlinear differential equations. Excercise 3.7 at page 67
Problem number : 14
Date solved : Thursday, October 02, 2025 at 07:15:47 PM
CAS classification : [_separable]

\begin{align*} x^{\prime }&=2 t \sqrt {x} \end{align*}

With initial conditions

\begin{align*} x \left (a \right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.036 (sec). Leaf size: 5

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

\[ x = 0 \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 6

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

\begin{align*} x(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.123 (sec). Leaf size: 20

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

\[ x{\left (t \right )} = \frac {a^{4}}{4} - \frac {a^{2} t^{2}}{2} + \frac {t^{4}}{4} \]