60.1.27 problem 27

Internal problem ID [10041]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 27
Date solved : Monday, January 27, 2025 at 06:19:16 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }+a y \left (y-x \right )-1&=0 \end{align*}

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 73

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

\[ y = \frac {2 a^{{3}/{2}} c_{1} x +\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right ) \sqrt {2}\, a x +2 \sqrt {a}\, {\mathrm e}^{-\frac {a \,x^{2}}{2}}}{a \left (\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right )+2 c_{1} \sqrt {a}\right )} \]

✓ Solution by Mathematica

Time used: 0.527 (sec). Leaf size: 134

DSolve[D[y[x],x] + a*y[x]*(y[x]-x) - 1==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\sqrt {2 \pi } c_1 x \text {erf}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )+\frac {2 \left (a x+c_1 e^{-\frac {a x^2}{2}}\right )}{\sqrt {a}}}{2 \sqrt {a}+\sqrt {2 \pi } c_1 \text {erf}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )} \\ y(x)\to \frac {\sqrt {\frac {2}{\pi }} e^{-\frac {a x^2}{2}}}{\sqrt {a} \text {erf}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )}+x \\ \end{align*}