30.5.5 problem 5

Internal problem ID [7504]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.6, Substitutions and Transformations. Exercises. page 76
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 04:40:32 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 21

ode:=t*diff(y(t),t)-y(t) = (t*y(t))^(1/2); 
dsolve(ode,y(t), singsol=all);
 

\[ -\frac {y}{\sqrt {t y}}+\frac {\ln \left (t \right )}{2}-c_1 = 0 \]

✓ Mathematica. Time used: 0.115 (sec). Leaf size: 17

ode=t*D[y[t],t]-y[t]==Sqrt[t*y[t]]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{4} t (\log (t)+c_1){}^2 \end{align*}

✓ Sympy. Time used: 0.301 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*Derivative(y(t), t) - sqrt(t*y(t)) - y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {C_{1}^{2} t}{4} + \frac {t \log {\left (t \right )}^{2}}{4} - \log {\left (t^{\frac {C_{1} t}{2}} \right )} \]