56.1.18 problem 18

Internal problem ID [8730]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 18
Date solved : Wednesday, March 05, 2025 at 06:14:10 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=\sqrt {\frac {1+y}{y^{2}}} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.442 (sec). Leaf size: 146

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

\[ y = -\frac {\left (1+i \sqrt {3}\right ) \left (-12 \sqrt {2}\, x +9 x^{2}+\sqrt {\left (-12 \sqrt {2}\, x +9 x^{2}-8\right ) \left (3 x -2 \sqrt {2}\right )^{2}}\right )^{{2}/{3}}-4 i \sqrt {3}-4 \left (-12 \sqrt {2}\, x +9 x^{2}+\sqrt {\left (-12 \sqrt {2}\, x +9 x^{2}-8\right ) \left (3 x -2 \sqrt {2}\right )^{2}}\right )^{{1}/{3}}+4}{4 \left (-12 \sqrt {2}\, x +9 x^{2}+\sqrt {\left (-12 \sqrt {2}\, x +9 x^{2}-8\right ) \left (3 x -2 \sqrt {2}\right )^{2}}\right )^{{1}/{3}}} \]

✓ Mathematica. Time used: 0.078 (sec). Leaf size: 123

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

\[ y(x)\to -\frac {1}{4} \left (1+i \sqrt {3}\right ) \sqrt [3]{9 x^2+\sqrt {81 x^4-216 \sqrt {2} x^3+288 x^2-64}-12 \sqrt {2} x}+\frac {i \left (\sqrt {3}+i\right )}{\sqrt [3]{9 x^2+\sqrt {81 x^4-216 \sqrt {2} x^3+288 x^2-64}-12 \sqrt {2} x}}+1 \]

✓ Sympy. Time used: 0.366 (sec). Leaf size: 31

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

\[ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {\frac {1 + \frac {1}{y}}{y}}}\, dy = x + \int \limits ^{1} \frac {1}{\sqrt {\frac {1}{y} + \frac {1}{y^{2}}}}\, dy \]