69.1.80 problem 125

Internal problem ID [14233]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 125
Date solved : Tuesday, January 28, 2025 at 06:22:49 AM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

\begin{align*} y \left (0\right )&=-1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.866 (sec). Leaf size: 24

dsolve([diff(y(x),x$2)^2+diff(y(x),x)^2=a^2,y(0) = -1, D(y)(0) = 0],y(x), singsol=all)
 

\begin{align*} y &= -a -1+a \cos \left (x \right ) \\ y &= a -1-a \cos \left (x \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 8.585 (sec). Leaf size: 37

DSolve[{(D[y[x],{x,2}])^2+(D[y[x],x])^2==a^2,{y[0]==-1,Derivative[1][y][0] ==0}},y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to a \left (\frac {1}{\sqrt {\sec ^2(x)}}-1\right )-1 \\ y(x)\to -\frac {a}{\sqrt {\sec ^2(x)}}+a-1 \\ \end{align*}