60.3.71 problem 1085

Internal problem ID [11067]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1085
Date solved : Thursday, March 13, 2025 at 08:23:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right ) y^{\prime }+\left (\frac {h^{\prime }\left (x \right ) \left (\frac {g^{\prime \prime }\left (x \right )}{g^{\prime }\left (x \right )}+\frac {\left (2 v -1\right ) g^{\prime }\left (x \right )}{g \left (x \right )}+\frac {2 h^{\prime }\left (x \right )}{h \left (x \right )}\right )}{h \left (x \right )}-\frac {h^{\prime \prime }\left (x \right )}{h \left (x \right )}+{g^{\prime }\left (x \right )}^{2}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 24

ode:=diff(diff(y(x),x),x)-(diff(diff(g(x),x),x)/diff(g(x),x)+(2*v-1)*diff(g(x),x)/g(x)+2*diff(h(x),x)/h(x))*diff(y(x),x)+(diff(h(x),x)/h(x)*(diff(diff(g(x),x),x)/diff(g(x),x)+(2*v-1)*diff(g(x),x)/g(x)+2*diff(h(x),x)/h(x))-diff(diff(h(x),x),x)/h(x)+diff(g(x),x)^2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = g \left (x \right )^{v} h \left (x \right ) \left (\operatorname {BesselJ}\left (v , g \left (x \right )\right ) c_{1} +\operatorname {BesselY}\left (v , g \left (x \right )\right ) c_{2} \right ) \]

✓ Mathematica. Time used: 0.136 (sec). Leaf size: 27

ode=-(D[y[x],x]*(((-1 + 2*v)*Derivative[1][g][x])/g[x] + (2*Derivative[1][h][x])/h[x] + Derivative[2][g][x]/Derivative[1][g][x])) + y[x]*(Derivative[1][g][x]^2 + (Derivative[1][h][x]*(((-1 + 2*v)*Derivative[1][g][x])/g[x] + (2*Derivative[1][h][x])/h[x] + Derivative[2][g][x]/Derivative[1][g][x]))/h[x] - Derivative[2][h][x]/h[x]) + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to h(x) g(x)^v (c_1 \operatorname {BesselJ}(v,g(x))+c_2 \operatorname {BesselY}(v,g(x))) \]

✗ Sympy

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

TypeError : cannot determine truth value of Relational