85.33.36 problem 36

Internal problem ID [22659]
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 65
Problem number : 36
Date solved : Thursday, October 02, 2025 at 09:03:04 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 20

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

\[ \frac {x^{2}}{2}-\frac {2 \sqrt {y}\, \left (y+3\right )}{3}+c_1 = 0 \]

✓ Mathematica. Time used: 3.036 (sec). Leaf size: 482

ode=(1+y[x])*D[y[x],{x,1}]==x*Sqrt[y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\sqrt [3]{9 x^4+36 c_1 x^2+3 \sqrt {\left (x^2+2 c_1\right ){}^2 \left (9 x^4+36 c_1 x^2+64+36 c_1{}^2\right )}+32+36 c_1{}^2}}{2\ 2^{2/3}}+\frac {2\ 2^{2/3}}{\sqrt [3]{9 x^4+36 c_1 x^2+3 \sqrt {\left (x^2+2 c_1\right ){}^2 \left (9 x^4+36 c_1 x^2+64+36 c_1{}^2\right )}+32+36 c_1{}^2}}-2\\ y(x)&\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{9 x^4+36 c_1 x^2+3 \sqrt {\left (x^2+2 c_1\right ){}^2 \left (9 x^4+36 c_1 x^2+64+36 c_1{}^2\right )}+32+36 c_1{}^2}}{4\ 2^{2/3}}-\frac {2^{2/3} \left (1+i \sqrt {3}\right )}{\sqrt [3]{9 x^4+36 c_1 x^2+3 \sqrt {\left (x^2+2 c_1\right ){}^2 \left (9 x^4+36 c_1 x^2+64+36 c_1{}^2\right )}+32+36 c_1{}^2}}-2\\ y(x)&\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{9 x^4+36 c_1 x^2+3 \sqrt {\left (x^2+2 c_1\right ){}^2 \left (9 x^4+36 c_1 x^2+64+36 c_1{}^2\right )}+32+36 c_1{}^2}}{4\ 2^{2/3}}+\frac {i 2^{2/3} \left (\sqrt {3}+i\right )}{\sqrt [3]{9 x^4+36 c_1 x^2+3 \sqrt {\left (x^2+2 c_1\right ){}^2 \left (9 x^4+36 c_1 x^2+64+36 c_1{}^2\right )}+32+36 c_1{}^2}}-2\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

Timed Out