Internal problem ID [14938]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 1. Basic concepts and definitions. Exercises page 18
Problem number: 6.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
\[ \boxed {y^{\prime }-\sqrt {x^{2}-y}=-x} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 171
dsolve(diff(y(x),x)=sqrt(x^2-y(x))-x,y(x), singsol=all)
\[ \frac {250 \left (x^{6} c_{1} y \left (x \right )^{2}+\frac {12 x^{4} c_{1} y \left (x \right )^{3}}{5}+\frac {48 x^{2} c_{1} y \left (x \right )^{4}}{25}+\frac {64 c_{1} y \left (x \right )^{5}}{125}-\frac {1}{125}\right ) \left (x^{2}-y \left (x \right )\right )^{\frac {3}{2}} \left (x^{2}+4 y \left (x \right )\right )-250 \left (x^{6} c_{1} y \left (x \right )^{2}+\frac {12 x^{4} c_{1} y \left (x \right )^{3}}{5}+\frac {48 x^{2} c_{1} y \left (x \right )^{4}}{25}+\frac {64 c_{1} y \left (x \right )^{5}}{125}+\frac {1}{125}\right ) \left (x^{4}+\frac {5 y \left (x \right ) x^{2}}{2}+10 y \left (x \right )^{2}\right ) x}{\left (5 x^{2}+4 y \left (x \right )\right )^{3} y \left (x \right )^{2} \left (-\sqrt {x^{2}-y \left (x \right )}+x \right )^{2} \left (3 x +2 \sqrt {x^{2}-y \left (x \right )}\right )^{3}} = 0 \]
✓ Solution by Mathematica
Time used: 4.748 (sec). Leaf size: 416
DSolve[y'[x]==Sqrt[x^2-y[x]]-x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {Root}\left [64 \text {$\#$1}^5+240 \text {$\#$1}^4 x^2+300 \text {$\#$1}^3 x^4+\text {$\#$1}^2 \left (125 x^6-40 e^{5 c_1} x\right )-10 \text {$\#$1} e^{5 c_1} x^3-4 e^{5 c_1} x^5+e^{10 c_1}\&,1\right ] \\ y(x)\to \text {Root}\left [64 \text {$\#$1}^5+240 \text {$\#$1}^4 x^2+300 \text {$\#$1}^3 x^4+\text {$\#$1}^2 \left (125 x^6-40 e^{5 c_1} x\right )-10 \text {$\#$1} e^{5 c_1} x^3-4 e^{5 c_1} x^5+e^{10 c_1}\&,2\right ] \\ y(x)\to \text {Root}\left [64 \text {$\#$1}^5+240 \text {$\#$1}^4 x^2+300 \text {$\#$1}^3 x^4+\text {$\#$1}^2 \left (125 x^6-40 e^{5 c_1} x\right )-10 \text {$\#$1} e^{5 c_1} x^3-4 e^{5 c_1} x^5+e^{10 c_1}\&,3\right ] \\ y(x)\to \text {Root}\left [64 \text {$\#$1}^5+240 \text {$\#$1}^4 x^2+300 \text {$\#$1}^3 x^4+\text {$\#$1}^2 \left (125 x^6-40 e^{5 c_1} x\right )-10 \text {$\#$1} e^{5 c_1} x^3-4 e^{5 c_1} x^5+e^{10 c_1}\&,4\right ] \\ y(x)\to \text {Root}\left [64 \text {$\#$1}^5+240 \text {$\#$1}^4 x^2+300 \text {$\#$1}^3 x^4+\text {$\#$1}^2 \left (125 x^6-40 e^{5 c_1} x\right )-10 \text {$\#$1} e^{5 c_1} x^3-4 e^{5 c_1} x^5+e^{10 c_1}\&,5\right ] \\ y(x)\to 0 \\ \end{align*}