71.8.45 problem 14 (e)

Internal problem ID [14479]
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 (e)
Date solved : Tuesday, January 28, 2025 at 06:41:46 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}{4}} \end{align*}

✓ Solution by Maple

Time used: 1.468 (sec). Leaf size: 17

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

\begin{align*} y &= -\frac {x^{2}}{4} \\ y &= \frac {1}{4}-\frac {x}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.444 (sec). Leaf size: 14

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

\[ y(x)\to \frac {1}{4} (1-2 x) \]