75.14.35 problem 361

Internal problem ID [16949]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 14. Differential equations admitting of depression of their order. Exercises page 107
Problem number : 361
Date solved : Tuesday, January 28, 2025 at 09:43:06 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }&={\mathrm e}^{2 y} \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.204 (sec). Leaf size: 12

dsolve([diff(y(x),x$2)=exp(2*y(x)),y(0) = 0, D(y)(0) = 1],y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.151 (sec). Leaf size: 13

DSolve[{D[y[x],{x,2}]==Exp[2*y[x]],{y[0]==0,Derivative[1][y][0] ==1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\log (1-x) \]