53.4.36 problem 39

Internal problem ID [8524]
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 : 39
Date solved : Monday, January 27, 2025 at 04:09:15 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} {y^{\prime \prime }}^{2}-2 y^{\prime \prime }+{y^{\prime }}^{2}-2 x y^{\prime }+x^{2}&=0 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.431 (sec). Leaf size: 23

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

\begin{align*} y &= \frac {\left (x +1\right )^{2}}{2} \\ y &= \frac {x^{2}}{2}+\sin \left (x \right )+\frac {1}{2} \\ \end{align*}

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

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

Not solved