85.9.5 problem 1 (e)

Internal problem ID [22479]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 37
Problem number : 1 (e)
Date solved : Thursday, October 02, 2025 at 08:40:43 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {x y^{2}+x}{4 y} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 35

ode:=diff(y(x),x) = 1/4*(x+x*y(x)^2)/y(x); 
ic:=[y(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\begin{align*} y &= \sqrt {{\mathrm e}^{\frac {\left (x -1\right ) \left (x +1\right )}{4}}-1} \\ y &= -\sqrt {{\mathrm e}^{\frac {\left (x -1\right ) \left (x +1\right )}{4}}-1} \\ \end{align*}

✓ Mathematica. Time used: 6.795 (sec). Leaf size: 45

ode=D[y[x],{x,1}]==(x+x*y[x]^2)/(4*y[x]); 
ic={y[1]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sqrt {e^{\frac {1}{4} \left (x^2-1\right )}-1}\\ y(x)&\to \sqrt {e^{\frac {1}{4} \left (x^2-1\right )}-1} \end{align*}

✓ Sympy. Time used: 0.441 (sec). Leaf size: 39

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

\[ \left [ y{\left (x \right )} = - \sqrt {\frac {e^{\frac {x^{2}}{4}}}{e^{\frac {1}{4}}} - 1}, \ y{\left (x \right )} = \sqrt {\frac {e^{\frac {x^{2}}{4}}}{e^{\frac {1}{4}}} - 1}\right ] \]