47.4.2 problem 2

Internal problem ID [9776]
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 : 2
Date solved : Tuesday, September 30, 2025 at 06:41:04 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x^{2} y^{\prime \prime }+{y^{\prime }}^{2}-2 x y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=5 \\ y^{\prime }\left (2\right )&=-4 \\ \end{align*}

✓ Maple. Time used: 0.082 (sec). Leaf size: 23

ode:=x^2*diff(diff(y(x),x),x)+diff(y(x),x)^2-2*x*diff(y(x),x) = 0; 
ic:=[y(2) = 5, D(y)(2) = -4]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {x^{2}}{2}+3 x +9 \ln \left (-3+x \right )-3-9 i \pi \]

✓ Mathematica. Time used: 11.238 (sec). Leaf size: 45

ode=x^2*D[y[x],{x,2}]+(D[y[x],x])^2-2*x*D[y[x],x]==0; 
ic={y[2]==5,Derivative[1][y][2]==-4}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x\frac {K[1]^2}{K[1]-3}dK[1]-\int _1^2\frac {K[1]^2}{K[1]-3}dK[1]+5 \end{align*}

✓ Sympy. Time used: 0.630 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - 2*x*Derivative(y(x), x) + Derivative(y(x), x)**2,0) 
ics = {y(2): 5, Subs(Derivative(y(x), x), x, 2): -4} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x^{2}}{2} + 3 x + 9 \log {\left (x - 3 \right )} - 3 - 9 i \pi \]