23.2.30 problem 31

Internal problem ID [5385]
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 : 31
Date solved : Tuesday, September 30, 2025 at 12:39:41 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 70

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

\begin{align*} y &= \frac {\left (a^{2}-4 b x \right )^{{3}/{2}}-6 b \left (a x -2 c_1 \right )}{12 b} \\ y &= \frac {\left (-a^{2}+4 b x \right ) \sqrt {a^{2}-4 b x}-6 b \left (a x -2 c_1 \right )}{12 b} \\ \end{align*}

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 68

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

\begin{align*} y(x)&\to -\frac {\left (a^2-4 b x\right )^{3/2}+6 a b x}{12 b}+c_1\\ y(x)&\to \frac {1}{2} \left (\frac {\left (a^2-4 b x\right )^{3/2}}{6 b}-a x\right )+c_1 \end{align*}

✓ Sympy. Time used: 0.176 (sec). Leaf size: 48

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

\[ \left [ y{\left (x \right )} = C_{1} - \frac {a x}{2} + \frac {\left (a^{2} - 4 b x\right )^{\frac {3}{2}}}{12 b}, \ y{\left (x \right )} = C_{1} - \frac {a x}{2} - \frac {\left (a^{2} - 4 b x\right )^{\frac {3}{2}}}{12 b}\right ] \]