7.5.65 problem 67

Internal problem ID [169]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.6 (substitution and exact equations). Problems at page 72
Problem number : 67
Date solved : Saturday, March 29, 2025 at 04:38:02 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} y&=x y^{\prime }-\frac {{y^{\prime }}^{2}}{4} \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 18

ode:=y(x) = x*diff(y(x),x)-1/4*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= x^{2} \\ y &= -\frac {c_1 \left (-4 x +c_1 \right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 25

ode=y[x]==x*D[y[x],x]-1/4*D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to c_1 x-\frac {c_1{}^2}{4} \\ y(x)\to x^2 \\ \end{align*}

✓ Sympy. Time used: 1.816 (sec). Leaf size: 10

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

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