47.4.32 problem 35

Internal problem ID [9806]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 35
Date solved : Tuesday, September 30, 2025 at 06:42:18 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 22

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

\[ y = \frac {x^{2}}{2}+\frac {c_1 \ln \left (x^{2}-c_1 \right )}{2}+c_2 \]

✓ Mathematica. Time used: 12.496 (sec). Leaf size: 52

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

\begin{align*} y(x)&\to \int _1^x\frac {K[1]^3}{K[1]^2+c_1}dK[1]+c_2\\ y(x)&\to c_2\\ y(x)&\to \frac {1}{2} \left (x^2-1+2 c_2\right ) \end{align*}

✓ Sympy. Time used: 0.632 (sec). Leaf size: 19

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

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