60.1.18 problem 18

Internal problem ID [10032]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 18
Date solved : Monday, January 27, 2025 at 06:18:52 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }-y^{2}-y x -x +1&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 62

dsolve(diff(y(x),x) - y(x)^2 - x*y(x) - x + 1=0,y(x), singsol=all)
 

\[ y = \frac {-i \sqrt {\pi }\, {\mathrm e}^{-2} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -2\right )}{2}\right )+2 \,{\mathrm e}^{\frac {x \left (x -4\right )}{2}}-2 c_{1}}{i \sqrt {\pi }\, {\mathrm e}^{-2} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -2\right )}{2}\right )+2 c_{1}} \]

✓ Solution by Mathematica

Time used: 0.318 (sec). Leaf size: 88

DSolve[D[y[x],x]- y[x]^2 - x*y[x] - x + 1==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -1+\frac {e^{\frac {1}{2} (x-4) x}}{-\int _1^xe^{\frac {1}{2} (K[1]-4) K[1]}dK[1]+c_1} \\ y(x)\to -1 \\ y(x)\to -\frac {e^{\frac {1}{2} (x-4) x}}{\int _1^xe^{\frac {1}{2} (K[1]-4) K[1]}dK[1]}-1 \\ \end{align*}