Internal problem ID [13301]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 4. SEPARABLE FIRST ORDER EQUATIONS. Additional exercises. page
90
Problem number: 4.3 (c).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {y^{\prime } x -\left (-y+x \right )^{2}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 67
dsolve(x*diff(y(x),x)=(x-y(x))^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sqrt {x}\, \left (\left (\operatorname {BesselK}\left (0, 2 \sqrt {x}\right ) c_{1} +\operatorname {BesselI}\left (0, 2 \sqrt {x}\right )\right ) \sqrt {x}+\operatorname {BesselK}\left (1, 2 \sqrt {x}\right ) c_{1} -\operatorname {BesselI}\left (1, 2 \sqrt {x}\right )\right )}{\operatorname {BesselK}\left (0, 2 \sqrt {x}\right ) c_{1} +\operatorname {BesselI}\left (0, 2 \sqrt {x}\right )} \]
✓ Solution by Mathematica
Time used: 0.251 (sec). Leaf size: 121
DSolve[x*y'[x]==(x-y[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x K_0\left (2 \sqrt {x}\right )+\sqrt {x} K_1\left (2 \sqrt {x}\right )+c_1 x \operatorname {BesselI}\left (0,2 \sqrt {x}\right )-c_1 \sqrt {x} \operatorname {BesselI}\left (1,2 \sqrt {x}\right )}{K_0\left (2 \sqrt {x}\right )+c_1 \operatorname {BesselI}\left (0,2 \sqrt {x}\right )} \\ y(x)\to x-\frac {\sqrt {x} \operatorname {BesselI}\left (1,2 \sqrt {x}\right )}{\operatorname {BesselI}\left (0,2 \sqrt {x}\right )} \\ \end{align*}