71.8.44 problem 14 (d)

Internal problem ID [14478]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.4.4, page 115
Problem number : 14 (d)
Date solved : Tuesday, January 28, 2025 at 06:41:43 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} y^{\prime }&=-\frac {x}{2}+\frac {\sqrt {x^{2}+4 y}}{2} \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 1.751 (sec). Leaf size: 69

dsolve([diff(y(x),x)=(-x+sqrt(x^2+4*y(x)))/2,y(1) = -1/5],y(x), singsol=all)
 

\begin{align*} y &= \frac {\left (x -1\right ) \sqrt {5}}{10}-\frac {x}{2}+\frac {3}{10} \\ y &= \frac {\left (-5+\sqrt {5}\right ) \left (-5+\sqrt {5}+10 x \right )}{100} \\ y &= -\frac {\left (2^{{1}/{3}} x -\frac {\left (50+20 \sqrt {5}\right )^{{1}/{3}}}{5}\right ) \left (50+20 \sqrt {5}\right )^{{1}/{3}} 2^{{1}/{3}}}{10} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.478 (sec). Leaf size: 51

DSolve[{D[y[x],x]==(-x+Sqrt[x^2+4*y[x]])/2,{y[1]==-2/10}},y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{100} \left (5+\sqrt {5}\right ) \left (-10 x+\sqrt {5}+5\right ) \\ y(x)\to \frac {1}{100} \left (\sqrt {5}-5\right ) \left (10 x+\sqrt {5}-5\right ) \\ \end{align*}