60.1.57 problem 58

Internal problem ID [10071]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 58
Date solved : Sunday, March 30, 2025 at 03:00:46 PM
CAS classification : [[_homogeneous, `class G`], _Chini]

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

✓ Maple. Time used: 0.045 (sec). Leaf size: 68

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

\[ -\frac {\ln \left (\sqrt {y}\, a x +b \,x^{2}-2 y\right )}{2}+\frac {a \sqrt {y}\, \operatorname {arctanh}\left (\frac {a \sqrt {y}+2 b x}{\sqrt {y \left (a^{2}+8 b \right )}}\right )}{\sqrt {y \left (a^{2}+8 b \right )}}+c_1 = 0 \]

✓ Mathematica. Time used: 0.269 (sec). Leaf size: 118

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

\[ \text {Solve}\left [\frac {a^2 \left (-\frac {2 a \text {arctanh}\left (\frac {a \left (1-\frac {4 b \sqrt {\frac {a^2 y(x)}{b^2 x^2}}}{a^2}\right )}{\sqrt {a^2+8 b}}\right )}{\sqrt {a^2+8 b}}-\log \left (a^2 \left (\sqrt {\frac {a^2 y(x)}{b^2 x^2}}+1\right )-\frac {2 a^2 y(x)}{b x^2}\right )\right )}{2 b}=\frac {a^2 \log (x)}{b}+c_1,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE -a*sqrt(y(x)) - b*x + Derivative(y(x), x) cannot be solved by the lie group method