78.7.8 problem 2 (a)

Internal problem ID [18194]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 11 (Reduction of order). Problems at page 87
Problem number : 2 (a)
Date solved : Tuesday, January 28, 2025 at 11:37:43 AM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 15

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

\begin{align*} y &= 1 \\ y &= -\frac {x^{3}}{3}+1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.146 (sec). Leaf size: 32

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

\begin{align*} y(x)\to \text {Indeterminate} \\ y(x)\to 0 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\text {ComplexInfinity}\right )-\frac {x^3}{6}+1 \\ \end{align*}