23.4.119 problem 119

Internal problem ID [6421]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 119
Date solved : Tuesday, September 30, 2025 at 02:56:29 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} \left (c \,x^{2}+2 b x +a \right )^{{3}/{2}} y^{\prime \prime }&=f \left (\frac {x}{\sqrt {c \,x^{2}+2 b x +a}}\right ) \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 43

ode:=(c*x^2+2*b*x+a)^(3/2)*diff(diff(y(x),x),x) = f(x/(c*x^2+2*b*x+a)^(1/2)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \int \int \frac {f \left (\frac {x}{\sqrt {c \,x^{2}+2 b x +a}}\right )}{\left (c \,x^{2}+2 b x +a \right )^{{3}/{2}}}d x d x +c_1 x +c_2 \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 66

ode=(a + 2*b*x + c*x^2)^(3/2)*D[y[x],{x,2}] == f[x/Sqrt[a + 2*b*x + c*x^2]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x\int _1^{K[2]}\frac {f\left (\frac {K[1]}{\sqrt {c K[1]^2+2 b K[1]+a}}\right )}{\left (c K[1]^2+2 b K[1]+a\right )^{3/2}}dK[1]dK[2]+c_2 x+c_1 \end{align*}

✓ Sympy. Time used: 4.036 (sec). Leaf size: 82

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq((a + 2*b*x + c*x**2)**(3/2)*Derivative(y(x), (x, 2)) - f(x/sqrt(a + 2*b*x + c*x**2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + C_{2} x + x \int \frac {f{\left (\frac {x}{\sqrt {a + 2 b x + c x^{2}}} \right )}}{\left (a + 2 b x + c x^{2}\right )^{\frac {3}{2}}}\, dx - \int \frac {x f{\left (\frac {x}{\sqrt {a + 2 b x + c x^{2}}} \right )}}{\left (a + 2 b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]