82.40.1 problem Ex. 1

Internal problem ID [18876]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. Exact differential equations, and equations of particular forms. Integration in series. problems at page 94
Problem number : Ex. 1
Date solved : Monday, March 31, 2025 at 06:17:53 PM
CAS classification : [[_3rd_order, _fully, _exact, _linear]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 79

ode:=x*diff(diff(diff(y(x),x),x),x)+(x^2-3)*diff(diff(y(x),x),x)+4*x*diff(y(x),x)+2*y(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: 0.147 (sec). Leaf size: 103

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*Derivative(y(x), x) + x*Derivative(y(x), (x, 3)) + (x**2 - 3)*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)) + Derivative(y(x), (x, 3))) - 2*y(x) + 3*Derivative(y(x), (x, 2)))/(4*x) cannot be solved by the factorable group method