75.14.12 problem 338

Internal problem ID [16926]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 14. Differential equations admitting of depression of their order. Exercises page 107
Problem number : 338
Date solved : Tuesday, January 28, 2025 at 09:41:06 AM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_poly_yn]]

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

With initial conditions

\begin{align*} y \left (1\right )&=\frac {\sqrt {2}}{5}\\ y^{\prime }\left (1\right )&=\frac {\sqrt {2}}{2} \end{align*}

✓ Solution by Maple

Time used: 0.128 (sec). Leaf size: 12

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

\[ y = \frac {x^{{5}/{2}} \sqrt {2}}{5} \]

✓ Solution by Mathematica

Time used: 0.088 (sec). Leaf size: 26

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

\[ y(x)\to \frac {1}{5} \sqrt {2} x^{3/2} \sqrt {x^2} \]