54.3.390 problem 1407

Internal problem ID [12685]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1407
Date solved : Friday, October 03, 2025 at 03:46:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\left (\frac {\left (1-\operatorname {al1} -\operatorname {bl1} \right ) \operatorname {b1}}{\operatorname {b1} x -\operatorname {a1}}+\frac {\left (1-\operatorname {al2} -\operatorname {bl2} \right ) \operatorname {b2}}{\operatorname {b2} x -\operatorname {a2}}+\frac {\left (1-\operatorname {al3} -\operatorname {bl3} \right ) \operatorname {b3}}{\operatorname {b3} x -\operatorname {a3}}\right ) y^{\prime }-\frac {\left (\frac {\operatorname {al1} \operatorname {bl1} \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right ) \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right )}{\operatorname {b1} x -\operatorname {a1}}+\frac {\operatorname {al2} \operatorname {bl2} \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right ) \left (\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1} \right )}{\operatorname {b2} x -\operatorname {a2}}+\frac {\operatorname {al3} \operatorname {bl3} \left (-\operatorname {a1} \operatorname {b3} +\operatorname {a3} \operatorname {b1} \right ) \left (\operatorname {a2} \operatorname {b3} -\operatorname {a3} \operatorname {b2} \right )}{\operatorname {b3} x -\operatorname {a3}}\right ) y}{\left (\operatorname {b1} x -\operatorname {a1} \right ) \left (\operatorname {b2} x -\operatorname {a2} \right ) \left (\operatorname {b3} x -\operatorname {a3} \right )} \end{align*}

✓ Maple. Time used: 0.793 (sec). Leaf size: 2220

ode:=diff(diff(y(x),x),x) = -((1-al1-bl1)*b1/(b1*x-a1)+(1-al2-bl2)*b2/(b2*x-a2)+(1-al3-bl3)*b3/(b3*x-a3))*diff(y(x),x)-1/(b1*x-a1)/(b2*x-a2)/(b3*x-a3)*(al1*bl1*(a1*b2-a2*b1)*(-a1*b3+a3*b1)/(b1*x-a1)+al2*bl2*(a2*b3-a3*b2)*(a1*b2-a2*b1)/(b2*x-a2)+al3*bl3*(-a1*b3+a3*b1)*(a2*b3-a3*b2)/(b3*x-a3))*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {Expression too large to display} \]

✓ Mathematica. Time used: 141.228 (sec). Leaf size: 1290

ode=D[y[x],{x,2}] == -((((al1*(-(a2*b1) + a1*b2)*(a3*b1 - a1*b3)*bl1)/(-a1 + b1*x) + (al2*(-(a2*b1) + a1*b2)*(-(a3*b2) + a2*b3)*bl2)/(-a2 + b2*x) + (al3*(a3*b1 - a1*b3)*(-(a3*b2) + a2*b3)*bl3)/(-a3 + b3*x))*y[x])/((-a1 + b1*x)*(-a2 + b2*x)*(-a3 + b3*x))) - ((b1*(1 - al1 - bl1))/(-a1 + b1*x) + (b2*(1 - al2 - bl2))/(-a2 + b2*x) + (b3*(1 - al3 - bl3))/(-a3 + b3*x))*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✗ Sympy

from sympy import * 
x = symbols("x") 
a1 = symbols("a1") 
a2 = symbols("a2") 
a3 = symbols("a3") 
al1 = symbols("al1") 
al2 = symbols("al2") 
al3 = symbols("al3") 
b1 = symbols("b1") 
b2 = symbols("b2") 
b3 = symbols("b3") 
bl1 = symbols("bl1") 
bl2 = symbols("bl2") 
bl3 = symbols("bl3") 
y = Function("y") 
ode = Eq((b1*(-al1 - bl1 + 1)/(-a1 + b1*x) + b2*(-al2 - bl2 + 1)/(-a2 + b2*x) + b3*(-al3 - bl3 + 1)/(-a3 + b3*x))*Derivative(y(x), x) + Derivative(y(x), (x, 2)) + (al1*bl1*(a1*b2 - a2*b1)*(-a1*b3 + a3*b1)/(-a1 + b1*x) + al2*bl2*(a1*b2 - a2*b1)*(a2*b3 - a3*b2)/(-a2 + b2*x) + al3*bl3*(-a1*b3 + a3*b1)*(a2*b3 - a3*b2)/(-a3 + b3*x))*y(x)/((-a1 + b1*x)*(-a2 + b2*x)*(-a3 + b3*x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out