23.3.214 problem 216

Internal problem ID [5928]
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 : 216
Date solved : Friday, October 03, 2025 at 01:45:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (\operatorname {b2} x +\operatorname {a2} \right ) y+\left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+x y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.139 (sec). Leaf size: 109

ode:=(b2*x+a2)*y(x)+(b1*x+a1)*diff(y(x),x)+x*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x \left (\sqrt {\operatorname {b1}^{2}-4 \operatorname {b2}}+\operatorname {b1} \right )}{2}} \left (\operatorname {KummerU}\left (\frac {\operatorname {a1} \sqrt {\operatorname {b1}^{2}-4 \operatorname {b2}}+\operatorname {a1} \operatorname {b1} -2 \operatorname {a2}}{2 \sqrt {\operatorname {b1}^{2}-4 \operatorname {b2}}}, \operatorname {a1} , \sqrt {\operatorname {b1}^{2}-4 \operatorname {b2}}\, x \right ) c_2 +\operatorname {KummerM}\left (\frac {\operatorname {a1} \sqrt {\operatorname {b1}^{2}-4 \operatorname {b2}}+\operatorname {a1} \operatorname {b1} -2 \operatorname {a2}}{2 \sqrt {\operatorname {b1}^{2}-4 \operatorname {b2}}}, \operatorname {a1} , \sqrt {\operatorname {b1}^{2}-4 \operatorname {b2}}\, x \right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 134

ode=(a2 + b2*x)*y[x] + (a1 + b1*x)*D[y[x],x] + x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-\frac {1}{2} x \left (\sqrt {\text {b1}^2-4 \text {b2}}+\text {b1}\right )} \left (c_1 \operatorname {HypergeometricU}\left (\frac {\text {a1} \left (\text {b1}+\sqrt {\text {b1}^2-4 \text {b2}}\right )-2 \text {a2}}{2 \sqrt {\text {b1}^2-4 \text {b2}}},\text {a1},\sqrt {\text {b1}^2-4 \text {b2}} x\right )+c_2 L_{\frac {2 \text {a2}-\text {a1} \left (\text {b1}+\sqrt {\text {b1}^2-4 \text {b2}}\right )}{2 \sqrt {\text {b1}^2-4 \text {b2}}}}^{\text {a1}-1}\left (\sqrt {\text {b1}^2-4 \text {b2}} x\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a1 = symbols("a1") 
a2 = symbols("a2") 
b1 = symbols("b1") 
b2 = symbols("b2") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (a1 + b1*x)*Derivative(y(x), x) + (a2 + b2*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None