30.7.1 problem 5 (d)

Internal problem ID [7576]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Review problem. (F) Clairaut equation. page 85
Problem number : 5 (d)
Date solved : Tuesday, September 30, 2025 at 04:54:25 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 19

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

\begin{align*} y &= -\frac {x^{2}}{8} \\ y &= c_1 \left (2 c_1 +x \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 25

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

\begin{align*} y(x)&\to c_1 (x+2 c_1)\\ y(x)&\to -\frac {x^2}{8} \end{align*}

✓ Sympy. Time used: 1.002 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = \frac {x^{2}}{8} - \frac {\left (C_{1} + x\right )^{2}}{8} \]