23.5.77 problem 77

Internal problem ID [6686]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 5. THE EQUATION IS LINEAR AND OF ORDER GREATER THAN TWO, page 410
Problem number : 77
Date solved : Tuesday, September 30, 2025 at 03:50:50 PM
CAS classification : [[_3rd_order, _fully, _exact, _linear]]

\begin{align*} 2 y+4 x y^{\prime }-\left (-x^{2}+3\right ) y^{\prime \prime }+x y^{\prime \prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 81

ode:=2*y(x)+4*x*diff(y(x),x)-(-x^2+3)*diff(diff(y(x),x),x)+x*diff(diff(diff(y(x),x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x^{2}}{2}} x^{5} c_1 \,\operatorname {Ei}_{1}\left (-\frac {x^{2}}{2}\right )-\frac {x^{5} \left (i \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right ) c_2 \sqrt {2}\, \sqrt {\pi }-30 c_3 \right ) {\mathrm e}^{-\frac {x^{2}}{2}}}{30}-\frac {c_2 \,x^{4}}{15}+2 c_1 \,x^{3}-\frac {c_2 \,x^{2}}{15}+4 c_1 x -\frac {c_2}{5} \]

✓ Mathematica. Time used: 0.069 (sec). Leaf size: 103

ode=2*y[x] + 4*x*D[y[x],x] - (3 - x^2)*D[y[x],{x,2}] + x*D[y[x],{x,3}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{240} e^{-\frac {x^2}{2}} \left (8 c_2 \left (\sqrt {2 \pi } x^5 \text {erfi}\left (\frac {x}{\sqrt {2}}\right )-2 e^{\frac {x^2}{2}} \left (x^4+x^2+3\right )\right )+15 c_3 x \left (x^4 \operatorname {ExpIntegralEi}\left (\frac {x^2}{2}\right )-2 e^{\frac {x^2}{2}} \left (x^2+2\right )\right )+240 c_1 x^5\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*Derivative(y(x), x) + x*Derivative(y(x), (x, 3)) - (3 - x**2)*Derivative(y(x), (x, 2)) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x*(x*Derivative(y(x), (x, 2)) + Derivati