74.18.65 problem 73

Internal problem ID [16543]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 73
Date solved : Thursday, March 13, 2025 at 08:17:31 AM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

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

With initial conditions

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

✓ Maple. Time used: 0.084 (sec). Leaf size: 11

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

\[ y = \sqrt {2 t -1} \]

✓ Mathematica. Time used: 0.371 (sec). Leaf size: 14

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

\[ y(t)\to \sqrt {2 t-1} \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(t), t) - (sqrt(-4*t**2*y(t)*Derivative(y(t), (t, 2)) + 4*t + y(t)**2) - y(t))/(2*t) cannot be solved by the factorable group method