54.3.424 problem 1445

Internal problem ID [12719]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1445
Date solved : Friday, October 03, 2025 at 03:47:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (2 f \left (x \right ) {g^{\prime }\left (x \right )}^{2} g \left (x \right )-\left (g \left (x \right )^{2}-1\right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )\right ) y^{\prime }}{f \left (x \right ) g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )}-\frac {\left (\left (g \left (x \right )^{2}-1\right ) \left (f^{\prime }\left (x \right ) \left (f \left (x \right ) g^{\prime \prime }\left (x \right )+2 f^{\prime }\left (x \right ) g^{\prime }\left (x \right )\right )-f \left (x \right ) f^{\prime \prime }\left (x \right ) g^{\prime }\left (x \right )\right )-\left (2 f^{\prime }\left (x \right ) g \left (x \right )+v \left (v +1\right ) f \left (x \right ) g^{\prime }\left (x \right )\right ) f \left (x \right ) {g^{\prime }\left (x \right )}^{2}\right ) y}{f \left (x \right )^{2} g^{\prime }\left (x \right ) \left (g \left (x \right )^{2}-1\right )} \end{align*}

✓ Maple. Time used: 0.073 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x) = -(2*f(x)*diff(g(x),x)^2*g(x)-(g(x)^2-1)*(f(x)*diff(diff(g(x),x),x)+2*diff(f(x),x)*diff(g(x),x)))/f(x)/diff(g(x),x)/(g(x)^2-1)*diff(y(x),x)-((g(x)^2-1)*(diff(f(x),x)*(f(x)*diff(diff(g(x),x),x)+2*diff(f(x),x)*diff(g(x),x))-f(x)*diff(diff(f(x),x),x)*diff(g(x),x))-(2*diff(f(x),x)*g(x)+v*(v+1)*f(x)*diff(g(x),x))*f(x)*diff(g(x),x)^2)/f(x)^2/diff(g(x),x)/(g(x)^2-1)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = f \left (x \right ) \left (\operatorname {LegendreP}\left (v , g \left (x \right )\right ) c_1 +\operatorname {LegendreQ}\left (v , g \left (x \right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.11 (sec). Leaf size: 23

ode=D[y[x],{x,2}] == -((D[y[x],x]*(2*f[x]*g[x]*Derivative[1][g][x]^2 - (-1 + g[x]^2)*(2*Derivative[1][f][x]*Derivative[1][g][x] + f[x]*Derivative[2][g][x])))/(f[x]*(-1 + g[x]^2)*Derivative[1][g][x])) - (y[x]*(-(f[x]*Derivative[1][g][x]^2*(2*g[x]*Derivative[1][f][x] + v*(1 + v)*f[x]*Derivative[1][g][x])) + (-1 + g[x]^2)*(-(f[x]*Derivative[1][g][x]*Derivative[2][f][x]) + Derivative[1][f][x]*(2*Derivative[1][f][x]*Derivative[1][g][x] + f[x]*Derivative[2][g][x]))))/(f[x]^2*(-1 + g[x]^2)*Derivative[1][g][x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to f(x) (c_1 \operatorname {LegendreP}(v,g(x))+c_2 \operatorname {LegendreQ}(v,g(x))) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
v = symbols("v") 
y = Function("y") 
f = Function("f") 
g = Function("g") 
ode = Eq((((f(x)*Derivative(g(x), (x, 2)) + 2*Derivative(f(x), x)*Derivative(g(x), x))*Derivative(f(x), x) - f(x)*Derivative(f(x), (x, 2))*Derivative(g(x), x))*(g(x)**2 - 1) - (v*(v + 1)*f(x)*Derivative(g(x), x) + 2*g(x)*Derivative(f(x), x))*f(x)*Derivative(g(x), x)**2)*y(x)/((g(x)**2 - 1)*f(x)**2*Derivative(g(x), x)) + ((-f(x)*Derivative(g(x), (x, 2)) - 2*Derivative(f(x), x)*Derivative(g(x), x))*(g(x)**2 - 1) + 2*f(x)*g(x)*Derivative(g(x), x)**2)*Derivative(y(x), x)/((g(x)**2 - 1)*f(x)*Derivative(g(x), x)) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : cannot determine truth value of Relational: x > _n + 2