23.3.123 problem 125

Internal problem ID [5837]
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 : 125
Date solved : Friday, October 03, 2025 at 01:44:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (\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: 0.009 (sec). Leaf size: 98

ode:=(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);
 

\[ y = {\mathrm e}^{-\frac {\operatorname {b1} x}{\operatorname {b0}}} \left (\operatorname {KummerU}\left (\frac {-\operatorname {a0} \operatorname {b0} \operatorname {b1} +\operatorname {a1} \,\operatorname {b0}^{2}+\operatorname {b1}^{2}}{2 \operatorname {b0}^{3}}, \frac {1}{2}, -\frac {\left (x \,\operatorname {b0}^{2}+\operatorname {a0} \operatorname {b0} -2 \operatorname {b1} \right )^{2}}{2 \operatorname {b0}^{3}}\right ) c_2 +\operatorname {KummerM}\left (\frac {-\operatorname {a0} \operatorname {b0} \operatorname {b1} +\operatorname {a1} \,\operatorname {b0}^{2}+\operatorname {b1}^{2}}{2 \operatorname {b0}^{3}}, \frac {1}{2}, -\frac {\left (x \,\operatorname {b0}^{2}+\operatorname {a0} \operatorname {b0} -2 \operatorname {b1} \right )^{2}}{2 \operatorname {b0}^{3}}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 132

ode=(a1 + b1*x)*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 e^{-\text {a0} x+\frac {\text {b1} x}{\text {b0}}-\frac {\text {b0} x^2}{2}} \left (c_2 \operatorname {Hypergeometric1F1}\left (\frac {\text {b0}^3-\text {a1} \text {b0}^2+\text {a0} \text {b1} \text {b0}-\text {b1}^2}{2 \text {b0}^3},\frac {1}{2},\frac {\left (x \text {b0}^2+\text {a0} \text {b0}-2 \text {b1}\right )^2}{2 \text {b0}^3}\right )+c_1 \operatorname {HermiteH}\left (\frac {-\text {b0}^3+\text {a1} \text {b0}^2-\text {a0} \text {b1} \text {b0}+\text {b1}^2}{\text {b0}^3},\frac {x \text {b0}^2+\text {a0} \text {b0}-2 \text {b1}}{\sqrt {2} \text {b0}^{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") 
y = Function("y") 
ode = Eq((a0 + b0*x)*Derivative(y(x), x) + (a1 + b1*x)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False