29.4.11 problem 100

Internal problem ID [4702]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 4
Problem number : 100
Date solved : Tuesday, March 04, 2025 at 07:06:13 PM
CAS classification : [[_homogeneous, `class G`], _Chini]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 68

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

\[ -\frac {\ln \left (\sqrt {y \left (x \right )}\, b x +a \,x^{2}-2 y \left (x \right )\right )}{2}+\frac {b \sqrt {y \left (x \right )}\, \operatorname {arctanh}\left (\frac {b \sqrt {y \left (x \right )}+2 a x}{\sqrt {y \left (x \right ) \left (b^{2}+8 a \right )}}\right )}{\sqrt {y \left (x \right ) \left (b^{2}+8 a \right )}}+c_{1} = 0 \]

✓ Mathematica. Time used: 0.276 (sec). Leaf size: 118

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

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

✗ Sympy

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

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