50.7.9 problem 2(b)

Internal problem ID [7913]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.9. Reduction of Order. Page 38
Problem number : 2(b)
Date solved : Monday, January 27, 2025 at 03:31:58 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _with_potential_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y y^{\prime \prime }&=y^{2} y^{\prime }+{y^{\prime }}^{2} \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: 0.139 (sec). Leaf size: 16

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

\[ y = -\frac {3}{8 \,{\mathrm e}^{\frac {3 x}{2}}-2} \]

✓ Solution by Mathematica

Time used: 1.773 (sec). Leaf size: 20

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

\[ y(x)\to \frac {3}{2-8 e^{3 x/2}} \]