69.1.14 problem 14

Internal problem ID [14088]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 14
Date solved : Wednesday, March 05, 2025 at 10:30:53 PM
CAS classification : [_separable]

\begin{align*} z-\left (-a^{2}+t^{2}\right ) z^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 26

ode:=z(t)-(-a^2+t^2)*diff(z(t),t) = 0; 
dsolve(ode,z(t), singsol=all);
 

\[ z = c_{1} \left (t -a \right )^{\frac {1}{2 a}} \left (t +a \right )^{-\frac {1}{2 a}} \]

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 34

ode=z[t]-(t^2-a^2)*D[z[t],t]==0; 
ic={}; 
DSolve[{ode,ic},z[t],t,IncludeSingularSolutions->True]
 

\begin{align*} z(t)\to c_1 \exp \left (\int _1^t\frac {1}{K[1]^2-a^2}dK[1]\right ) \\ z(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.470 (sec). Leaf size: 19

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

\[ z{\left (t \right )} = C_{1} e^{\frac {\log {\left (- a + t \right )} - \log {\left (a + t \right )}}{2 a}} \]