23.2.123 problem 125

Internal problem ID [5478]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 125
Date solved : Tuesday, September 30, 2025 at 12:45:11 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 205

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

\begin{align*} \frac {-c_1 \left (y a -\sqrt {a^{2} y^{2}-4 x^{2} b}\right ) {\left (\frac {a \left (-y \left (a +1\right ) \sqrt {a^{2} y^{2}-4 x^{2} b}+\left (a^{2}+a \right ) y^{2}-2 x^{2} b \right )}{2 x^{2}}\right )}^{\frac {-a -2}{2 a +2}}+x^{2}}{x} &= 0 \\ \frac {c_1 2^{\frac {a +2}{2 a +2}} \left (y a +\sqrt {a^{2} y^{2}-4 x^{2} b}\right ) {\left (\frac {a \left (y \left (a +1\right ) \sqrt {a^{2} y^{2}-4 x^{2} b}+\left (a^{2}+a \right ) y^{2}-2 x^{2} b \right )}{x^{2}}\right )}^{\frac {-a -2}{2 a +2}}+x^{2}}{x} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 1.43 (sec). Leaf size: 423

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

\begin{align*} \text {Solve}\left [-\frac {i \left (2 \log \left (-i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}+2 i \sqrt {b}\right )+2 (a+1) \log \left (i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}-2 i \sqrt {b}\right )-(a+2) \log \left (\frac {i (a+2) y(x) \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}{x}+2 \sqrt {b} \left (\sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}-\frac {i (a+2) y(x)}{x}\right )+\frac {a^2 y(x)^2}{x^2}-4 b\right )\right )}{4 (a+1)}=c_1-\frac {1}{2} i \log (x),y(x)\right ]\\ \text {Solve}\left [\frac {i \left (2 (a+1) \log \left (-i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}+2 i \sqrt {b}\right )+2 \log \left (i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}-2 i \sqrt {b}\right )-(a+2) \log \left (-\frac {i (a+2) y(x) \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}{x}+2 \sqrt {b} \left (\sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {i (a+2) y(x)}{x}\right )+\frac {a^2 y(x)^2}{x^2}-4 b\right )\right )}{4 (a+1)}=\frac {1}{2} i \log (x)+c_1,y(x)\right ] \end{align*}

✓ Sympy. Time used: 14.652 (sec). Leaf size: 141

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

\[ \left [ y{\left (x \right )} = C_{1} e^{- a \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1} \left (a + \sqrt {- 4 u_{1}^{2} b + a^{2}} + 2\right )}\, du_{1} - \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {- 4 u_{1}^{2} b + a^{2}}}{u_{1} \left (a + \sqrt {- 4 u_{1}^{2} b + a^{2}} + 2\right )}\, du_{1}}, \ y{\left (x \right )} = C_{1} e^{- a \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1} \left (a - \sqrt {- 4 u_{1}^{2} b + a^{2}} + 2\right )}\, du_{1} + \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {- 4 u_{1}^{2} b + a^{2}}}{u_{1} \left (a - \sqrt {- 4 u_{1}^{2} b + a^{2}} + 2\right )}\, du_{1}}\right ] \]