53.4.31 problem 34

Internal problem ID [8519]
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 : 34
Date solved : Monday, January 27, 2025 at 04:09:10 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x^{2} y^{\prime \prime }&=y^{\prime } \left (2 x -y^{\prime }\right ) \end{align*}

With initial conditions

\begin{align*} y \left (-1\right )&=5\\ y^{\prime }\left (-1\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.088 (sec). Leaf size: 20

dsolve([x^2*diff(y(x),x$2)=diff(y(x),x)*(2*x-diff(y(x),x)),y(-1) = 5, D(y)(-1) = 1],y(x), singsol=all)
 

\[ y = \frac {x^{2}}{2}-2 x +4 \ln \left (x +2\right )+\frac {5}{2} \]

✓ Solution by Mathematica

Time used: 0.481 (sec). Leaf size: 23

DSolve[{x^2*D[y[x],{x,2}]==D[y[x],x]*(2*x-D[y[x],x]),{y[-1]==5,Derivative[1][y][-1]==1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} \left (x^2-4 x+8 \log (x+2)+5\right ) \]