53.4.16 problem 17

Internal problem ID [8504]
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 : 17
Date solved : Monday, January 27, 2025 at 04:07:36 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&={\frac {1}{2}} \end{align*}

✓ Solution by Maple

Time used: 0.072 (sec). Leaf size: 10

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

\[ y = \operatorname {arctanh}\left (\frac {x}{2}\right )+1 \]

✓ Solution by Mathematica

Time used: 0.287 (sec). Leaf size: 13

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

\[ y(x)\to \text {arctanh}\left (\frac {x}{2}\right )+1 \]