67.8.5 problem 13.1 (e)

Internal problem ID [16500]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.1 (e)
Date solved : Thursday, October 02, 2025 at 01:35:37 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 14

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

\[ y = c_1 +{\mathrm e}^{-x^{2}} c_2 \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 21

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

\begin{align*} y(x)&\to c_2-\frac {1}{2} c_1 e^{-x^2} \end{align*}

✓ Sympy. Time used: 0.123 (sec). Leaf size: 53

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

\[ y{\left (x \right )} = C_{1} + x^{- 2 \left (\operatorname {re}{\left (x\right )}\right )^{2} + 2 \left (\operatorname {im}{\left (x\right )}\right )^{2} + 2} \left (C_{2} \sin {\left (4 \log {\left (x \right )} \left |{\operatorname {re}{\left (x\right )} \operatorname {im}{\left (x\right )}}\right | \right )} + C_{3} \cos {\left (4 \log {\left (x \right )} \operatorname {re}{\left (x\right )} \operatorname {im}{\left (x\right )} \right )}\right ) \]