56.33.3 problem Ex 3

Internal problem ID [14271]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 57. Dependent variable absent. Page 132
Problem number : Ex 3
Date solved : Thursday, October 02, 2025 at 09:27:30 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 23

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

\[ y = \frac {c_1 \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )}{2}+x +c_2 \]

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 29

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

\begin{align*} y(x)&\to \sqrt {\frac {\pi }{2}} c_1 \text {erf}\left (\frac {x}{\sqrt {2}}\right )+x+c_2 \end{align*}

✓ Sympy. Time used: 1.615 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = C_{1} + C_{2} \operatorname {erf}{\left (\frac {\sqrt {2} x}{2} \right )} + x \]