53.4.23 problem 25

Internal problem ID [8511]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 25
Date solved : Monday, January 27, 2025 at 04:08:38 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 24

dsolve(diff(y(x),x$2)=exp(x)*diff(y(x),x)^2,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.929 (sec). Leaf size: 37

DSolve[D[y[x],{x,2}]==Exp[x](D[y[x],x])^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {-x+\log \left (e^x+c_1\right )+c_1 c_2}{c_1} \\ y(x)\to \text {Indeterminate} \\ y(x)\to c_2 \\ \end{align*}