83.36.7 problem 7

Internal problem ID [19370]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise VIII (A) at page 125
Problem number : 7
Date solved : Monday, March 31, 2025 at 07:11:41 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-x y^{\prime \prime }-y^{\prime }+x y&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 48

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

\[ y = \left (c_3 \,\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -1\right )}{2}\right ) {\mathrm e}^{-\frac {1}{2}+2 x}+c_1 \,{\mathrm e}^{2 x}-\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +1\right )}{2}\right ) {\mathrm e}^{-\frac {1}{2}} c_3 +c_2 \right ) {\mathrm e}^{-x} \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 70

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

\[ y(x)\to \frac {1}{2} e^{-x} \left (\sqrt {\frac {2 \pi }{e}} c_3 \left (e^{2 x} \text {erfi}\left (\frac {x-1}{\sqrt {2}}\right )-\text {erfi}\left (\frac {x+1}{\sqrt {2}}\right )\right )+2 c_2 e^{2 x}+2 c_1\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE -x*y(x) + x*Derivative(y(x), (x, 2)) + Derivative(y(x), x) - Derivative(y(x), (x, 3)) cannot be solved by the factorable group method