60.1.205 problem 209

Internal problem ID [10219]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 209
Date solved : Sunday, March 30, 2025 at 03:32:11 PM
CAS classification : [_quadrature]

\begin{align*} y y^{\prime }-\sqrt {a y^{2}+b}&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.483 (sec). Leaf size: 94

ode=y[x]*D[y[x],x]-Sqrt[a*y[x]^2+b]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\sqrt {-b+a^2 (x+c_1){}^2}}{\sqrt {a}} \\ y(x)\to \frac {\sqrt {-b+a^2 (x+c_1){}^2}}{\sqrt {a}} \\ y(x)\to -\frac {i \sqrt {b}}{\sqrt {a}} \\ y(x)\to \frac {i \sqrt {b}}{\sqrt {a}} \\ \end{align*}

✓ Sympy. Time used: 1.556 (sec). Leaf size: 109

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

\[ \left [ y{\left (x \right )} = \begin {cases} - \sqrt {2} \sqrt {C_{1} \sqrt {b} + \sqrt {b} x} & \text {for}\: a = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \sqrt {2} \sqrt {C_{1} \sqrt {b} + \sqrt {b} x} & \text {for}\: a = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} - \sqrt {C_{1}^{2} a + 2 C_{1} a x + a x^{2} - \frac {b}{a}} & \text {for}\: a \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \sqrt {C_{1}^{2} a + 2 C_{1} a x + a x^{2} - \frac {b}{a}} & \text {for}\: a \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}\right ] \]