23.5.109 problem 109

Internal problem ID [6718]
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 : 109
Date solved : Friday, October 03, 2025 at 02:09:48 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.041 (sec). Leaf size: 123

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

\[ y = \frac {\left (3 i \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{-\frac {1}{2}} \int x \left (-\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (-2+\ln \left (x \right )\right )}{2}\right ) {\mathrm e}^{-\frac {3}{2}} x +\operatorname {erf}\left (\frac {i \sqrt {2}\, \left (-1+\ln \left (x \right )\right )}{2}\right )\right )d x +\left (i x^{3} \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{-2}+3 c_2 \right ) \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (-2+\ln \left (x \right )\right )}{2}\right )+\left (-\frac {3 i x^{2} \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{-\frac {1}{2}}}{2}+3 c_3 \right ) \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (-1+\ln \left (x \right )\right )}{2}\right )+3 c_1 \right ) {\mathrm e}^{-\frac {\ln \left (x \right )^{2}}{2}} x^{2}}{3} \]

✓ Mathematica. Time used: 0.075 (sec). Leaf size: 105

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

\begin{align*} y(x)&\to \frac {1}{6} x^2 e^{\frac {1}{2} \left (-\log ^2(x)-5\right )} \left (3 \sqrt {2 e \pi } c_2 \text {erfi}\left (\frac {\log (x)-2}{\sqrt {2}}\right )+e^2 \left (\sqrt {2 \pi } \text {erfi}\left (\frac {\log (x)+1}{\sqrt {2}}\right )+3 \sqrt {2 \pi } c_3 \text {erfi}\left (\frac {\log (x)-1}{\sqrt {2}}\right )+6 \sqrt {e} c_1\right )\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x**2*(-x*Derivative(y(x), (x, 3)) + 2*x -