23.3.332 problem 334

Internal problem ID [6046]
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 : 334
Date solved : Friday, October 03, 2025 at 01:45:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (\operatorname {c2} \,x^{2}+\operatorname {b2} x +\operatorname {a2} \right ) y+x \left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.146 (sec). Leaf size: 110

ode:=(c2*x^2+b2*x+a2)*y(x)+x*(b1*x+a1)*diff(y(x),x)+x^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {\operatorname {b1} x}{2}} x^{-\frac {\operatorname {a1}}{2}} \left (\operatorname {WhittakerW}\left (-\frac {\operatorname {a1} \operatorname {b1} -2 \operatorname {b2}}{2 \sqrt {\operatorname {b1}^{2}-4 \operatorname {c2}}}, \frac {\sqrt {\operatorname {a1}^{2}-2 \operatorname {a1} -4 \operatorname {a2} +1}}{2}, \sqrt {\operatorname {b1}^{2}-4 \operatorname {c2}}\, x \right ) c_2 +\operatorname {WhittakerM}\left (-\frac {\operatorname {a1} \operatorname {b1} -2 \operatorname {b2}}{2 \sqrt {\operatorname {b1}^{2}-4 \operatorname {c2}}}, \frac {\sqrt {\operatorname {a1}^{2}-2 \operatorname {a1} -4 \operatorname {a2} +1}}{2}, \sqrt {\operatorname {b1}^{2}-4 \operatorname {c2}}\, x \right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.089 (sec). Leaf size: 223

ode=(a2 + b2*x + c2*x^2)*y[x] + x*(a1 + b1*x)*D[y[x],x] + x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x^{\frac {1}{2} \left (\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}-\text {a1}+1\right )} e^{-\frac {1}{2} x \left (\sqrt {\text {b1}^2-4 \text {c2}}+\text {b1}\right )} \left (c_1 \operatorname {HypergeometricU}\left (\frac {\sqrt {\text {b1}^2-4 \text {c2}} \left (\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}+1\right )+\text {a1} \text {b1}-2 \text {b2}}{2 \sqrt {\text {b1}^2-4 \text {c2}}},\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}+1,\sqrt {\text {b1}^2-4 \text {c2}} x\right )+c_2 L_{\frac {-\sqrt {\text {b1}^2-4 \text {c2}} \left (\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}+1\right )-\text {a1} \text {b1}+2 \text {b2}}{2 \sqrt {\text {b1}^2-4 \text {c2}}}}^{\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}}\left (\sqrt {\text {b1}^2-4 \text {c2}} x\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a1 = symbols("a1") 
a2 = symbols("a2") 
b1 = symbols("b1") 
b2 = symbols("b2") 
c2 = symbols("c2") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(a1 + b1*x)*Derivative(y(x), x) + (a2 + b2*x + c2*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None