23.4.263 problem 263

Internal problem ID [6565]
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 : 263
Date solved : Tuesday, September 30, 2025 at 03:04:33 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} \left (-\left (1-y\right ) \left (a -y\right )+\left (1-y\right ) y+\left (a -y\right ) y\right ) {y^{\prime }}^{2}+2 \left (1-y\right ) \left (a -y\right ) y y^{\prime \prime }&=\operatorname {a3} \left (1-y\right )^{2} \left (a -y\right )^{2}+\operatorname {a1} \left (1-y\right )^{2} y^{2}+\operatorname {a2} \left (a -y\right )^{2} y^{2}+\operatorname {a0} \left (a -y\right )^{2} y^{2} \left (1-y^{2}\right ) \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 281

ode:=(-(1-y(x))*(a-y(x))+(1-y(x))*y(x)+(a-y(x))*y(x))*diff(y(x),x)^2+2*(1-y(x))*(a-y(x))*y(x)*diff(diff(y(x),x),x) = a3*(1-y(x))^2*(a-y(x))^2+a1*(1-y(x))^2*y(x)^2+a2*(a-y(x))^2*y(x)^2+a0*(a-y(x))^2*y(x)^2*(1-y(x)^2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \int _{}^{y}\frac {1}{\sqrt {-2 \ln \left (-1+\textit {\_a} \right ) \textit {\_a}^{3} \operatorname {a0} +2 \ln \left (-1+\textit {\_a} \right ) \textit {\_a}^{2} a \operatorname {a0} -\textit {\_a}^{4} \operatorname {a0} +\textit {\_a}^{3} a \operatorname {a0} +2 \ln \left (-1+\textit {\_a} \right ) \textit {\_a}^{2} \operatorname {a0} -2 \ln \left (-1+\textit {\_a} \right ) \textit {\_a} a \operatorname {a0} +\textit {\_a}^{3} c_1 -c_1 \,\textit {\_a}^{2} a +\textit {\_a}^{3} \operatorname {a0} -\textit {\_a}^{2} a \operatorname {a0} -c_1 \,\textit {\_a}^{2}+c_1 \textit {\_a} a -\operatorname {a1} \,\textit {\_a}^{2}-\operatorname {a2} \,\textit {\_a}^{2}-\textit {\_a}^{2} \operatorname {a3} +\operatorname {a2} \textit {\_a} a +\textit {\_a} a \operatorname {a3} +\textit {\_a} \operatorname {a1} +\textit {\_a} \operatorname {a3} -a \operatorname {a3}}}d \textit {\_a} -x -c_2 &= 0 \\ -\int _{}^{y}\frac {1}{\sqrt {-2 \ln \left (-1+\textit {\_a} \right ) \textit {\_a}^{3} \operatorname {a0} +2 \ln \left (-1+\textit {\_a} \right ) \textit {\_a}^{2} a \operatorname {a0} -\textit {\_a}^{4} \operatorname {a0} +\textit {\_a}^{3} a \operatorname {a0} +2 \ln \left (-1+\textit {\_a} \right ) \textit {\_a}^{2} \operatorname {a0} -2 \ln \left (-1+\textit {\_a} \right ) \textit {\_a} a \operatorname {a0} +\textit {\_a}^{3} c_1 -c_1 \,\textit {\_a}^{2} a +\textit {\_a}^{3} \operatorname {a0} -\textit {\_a}^{2} a \operatorname {a0} -c_1 \,\textit {\_a}^{2}+c_1 \textit {\_a} a -\operatorname {a1} \,\textit {\_a}^{2}-\operatorname {a2} \,\textit {\_a}^{2}-\textit {\_a}^{2} \operatorname {a3} +\operatorname {a2} \textit {\_a} a +\textit {\_a} a \operatorname {a3} +\textit {\_a} \operatorname {a1} +\textit {\_a} \operatorname {a3} -a \operatorname {a3}}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 22.876 (sec). Leaf size: 1661

ode=(-((1 - y[x])*(a - y[x])) + (1 - y[x])*y[x] + (a - y[x])*y[x])*D[y[x],x]^2 + 2*(1 - y[x])*(a - y[x])*y[x]*D[y[x],{x,2}] == a3*(1 - y[x])^2*(a - y[x])^2 + a1*(1 - y[x])^2*y[x]^2 + a2*(a - y[x])^2*y[x]^2 + a0*(a - y[x])^2*y[x]^2*(1 - y[x]^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
a0 = symbols("a0") 
a1 = symbols("a1") 
a2 = symbols("a2") 
a3 = symbols("a3") 
y = Function("y") 
ode = Eq(-a0*(1 - y(x)**2)*(a - y(x))**2*y(x)**2 - a1*(1 - y(x))**2*y(x)**2 - a2*(a - y(x))**2*y(x)**2 - a3*(1 - y(x))**2*(a - y(x))**2 + (2 - 2*y(x))*(a - y(x))*y(x)*Derivative(y(x), (x, 2)) + ((1 - y(x))*y(x) + (a - y(x))*(y(x) - 1) + (a - y(x))*y(x))*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt((-a**2*a0*y(x)**4 + a**2*a0*y(x)**2 + a**2*a2*y(x)**2 + a*