34.7.8 problem 17

Internal problem ID [7985]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 12. Linear equations of order n. Supplemetary problems. Page 81
Problem number : 17
Date solved : Tuesday, September 30, 2025 at 05:13:52 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }+{y^{\prime }}^{2}+1&=0 \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 15

ode:=diff(diff(y(x),x),x)+diff(y(x),x)^2+1 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \ln \left (-c_1 \sin \left (x \right )+c_2 \cos \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.423 (sec). Leaf size: 40

ode=D[y[x],{x,2}]+D[y[x],x]^2+1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.727 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x)**2 + Derivative(y(x), (x, 2)) + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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