1.5.50 problem 50

Internal problem ID [154]
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 : 50
Date solved : Tuesday, September 30, 2025 at 03:47:22 AM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 2.282 (sec). Leaf size: 23

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

\begin{align*} y(x)&\to -\frac {x^2}{2}-\log (\cos (x+c_1))+c_2 \end{align*}

✓ Sympy. Time used: 0.891 (sec). Leaf size: 22

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

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