60.1.62 problem 63

Internal problem ID [10076]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 63
Date solved : Wednesday, March 05, 2025 at 08:23:16 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }-\frac {1+y^{2}}{{| y+\sqrt {1+y}|} \left (1+x \right )^{{3}/{2}}}&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 35

ode:=diff(y(x),x)-(1+y(x)^2)/abs(y(x)+(1+y(x))^(1/2))/(1+x)^(3/2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ -\frac {2}{\sqrt {x +1}}-\int _{}^{y}\frac {{| \textit {\_a} +\sqrt {\textit {\_a} +1}|}}{\textit {\_a}^{2}+1}d \textit {\_a} +c_{1} = 0 \]

✓ Mathematica. Time used: 0.379 (sec). Leaf size: 62

ode=D[y[x],x] - (1+ y[x]^2)/(Abs[y[x]+Sqrt[1+y[x]]]*Sqrt[1+x]^3)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\left | K[1]+\sqrt {K[1]+1}\right | }{K[1]^2+1}dK[1]\&\right ]\left [-\frac {2}{\sqrt {x+1}}+c_1\right ] \\ y(x)\to -i \\ y(x)\to i \\ \end{align*}

✓ Sympy. Time used: 4.151 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (y(x)**2 + 1)/((x + 1)**(3/2)*Abs(sqrt(y(x) + 1) + y(x))),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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