68.3.6 problem 6

Internal problem ID [17153]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.1, page 32
Problem number : 6
Date solved : Thursday, October 02, 2025 at 01:45:19 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=4 t^{2}-t y^{2} \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.207 (sec). Leaf size: 127

ode:=diff(y(t),t) = 4*t^2-t*y(t)^2; 
ic:=[y(2) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -\frac {2 \left (\left (2 \sqrt {2}\, \operatorname {BesselK}\left (\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right )+\operatorname {BesselK}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )\right ) \operatorname {BesselI}\left (-\frac {3}{5}, \frac {4 t^{{5}/{2}}}{5}\right )+\operatorname {BesselK}\left (\frac {3}{5}, \frac {4 t^{{5}/{2}}}{5}\right ) \left (-2 \operatorname {BesselI}\left (-\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right ) \sqrt {2}+\operatorname {BesselI}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )\right )\right ) \sqrt {t}}{\left (-2 \sqrt {2}\, \operatorname {BesselK}\left (\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right )-\operatorname {BesselK}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )\right ) \operatorname {BesselI}\left (\frac {2}{5}, \frac {4 t^{{5}/{2}}}{5}\right )+\operatorname {BesselK}\left (\frac {2}{5}, \frac {4 t^{{5}/{2}}}{5}\right ) \left (-2 \operatorname {BesselI}\left (-\frac {3}{5}, \frac {16 \sqrt {2}}{5}\right ) \sqrt {2}+\operatorname {BesselI}\left (\frac {2}{5}, \frac {16 \sqrt {2}}{5}\right )\right )} \]

✓ Mathematica. Time used: 0.161 (sec). Leaf size: 709

ode=D[y[t],t]==4*t^2-t*y[t]^2; 
ic={y[2]==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-4*t**2 + t*y(t)**2 + Derivative(y(t), t),0) 
ics = {y(2): 1} 
dsolve(ode,func=y(t),ics=ics)
 

ValueError : Rational Solution doesnt exist