23.3.424 problem 429

Internal problem ID [6138]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 429
Date solved : Tuesday, September 30, 2025 at 02:22:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y-x y^{\prime }+2 x^{2} y^{\prime \prime }&=x^{2} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 18

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

\[ y = c_2 x +\sqrt {x}\, c_1 +\frac {x^{2}}{3} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 25

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

\begin{align*} y(x)&\to \frac {x^2}{3}+c_2 x+c_1 \sqrt {x} \end{align*}

✓ Sympy. Time used: 0.202 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = C_{1} \sqrt {x} + C_{2} x + \frac {x^{2}}{3} \]