53.4.9 problem 10

Internal problem ID [8497]
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 : 10
Date solved : Monday, January 27, 2025 at 04:07:25 PM
CAS classification : [[_2nd_order, _missing_y]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 16

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

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

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 19

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

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