Internal problem ID [12177]
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.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {{y^{\prime \prime }}^{2}+{y^{\prime }}^{2}=a^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.141 (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 \left (x \right ) = -1-a +a \cos \left (x \right ) y \left (x \right ) = a -1-a \cos \left (x \right ) \end{align*}
✓ Solution by Mathematica
Time used: 15.637 (sec). Leaf size: 37
DSolve[{(y''[x])^2+(y'[x])^2==a^2,{y[0]==-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*}