23.3.124 problem 126

Internal problem ID [5838]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 126
Date solved : Friday, October 03, 2025 at 01:44:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (\operatorname {c1} \,x^{2}+\operatorname {b1} x +\operatorname {a1} \right ) y+\left (\operatorname {b0} x +\operatorname {a0} \right ) y^{\prime }+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 280

ode:=(c1*x^2+b1*x+a1)*y(x)+(b0*x+a0)*diff(y(x),x)+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.133 (sec). Leaf size: 303

ode=(a1 + b1*x + c1*x^2)*y[x] + (a0 + b0*x)*D[y[x],x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \exp \left (-\frac {x \left (2 \text {a0} \left (\sqrt {\text {b0}^2-4 \text {c1}}+\text {b0}\right )+x \left (\text {b0} \sqrt {\text {b0}^2-4 \text {c1}}+\text {b0}^2-4 \text {c1}\right )-4 \text {b1}\right )}{4 \sqrt {\text {b0}^2-4 \text {c1}}}\right ) \left (c_1 \operatorname {HermiteH}\left (\frac {-\text {b0}^3-\sqrt {\text {b0}^2-4 \text {c1}} \text {b0}^2-2 \text {a0} \text {b1} \text {b0}+4 \text {c1} \text {b0}+2 \text {b1}^2+2 \text {a1} \left (\text {b0}^2-4 \text {c1}\right )+2 \text {a0}^2 \text {c1}+4 \sqrt {\text {b0}^2-4 \text {c1}} \text {c1}}{2 \left (\text {b0}^2-4 \text {c1}\right )^{3/2}},\frac {x \text {b0}^2+\text {a0} \text {b0}-2 \text {b1}-4 \text {c1} x}{\sqrt {2} \left (\text {b0}^2-4 \text {c1}\right )^{3/4}}\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {\text {b0}^3-2 \text {a1} \text {b0}^2+\sqrt {\text {b0}^2-4 \text {c1}} \text {b0}^2+2 \text {a0} \text {b1} \text {b0}-4 \text {c1} \text {b0}-2 \text {b1}^2-2 \text {a0}^2 \text {c1}+8 \text {a1} \text {c1}-4 \sqrt {\text {b0}^2-4 \text {c1}} \text {c1}}{4 \left (\text {b0}^2-4 \text {c1}\right )^{3/2}},\frac {1}{2},\frac {\left (x \text {b0}^2+\text {a0} \text {b0}-2 \text {b1}-4 \text {c1} x\right )^2}{2 \left (\text {b0}^2-4 \text {c1}\right )^{3/2}}\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a0 = symbols("a0") 
a1 = symbols("a1") 
b0 = symbols("b0") 
b1 = symbols("b1") 
c1 = symbols("c1") 
y = Function("y") 
ode = Eq((a0 + b0*x)*Derivative(y(x), x) + (a1 + b1*x + c1*x**2)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False