Internal problem ID [10156]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter 2, differential equations of the first order and the first degree. Article 18.
Transformation of variables. Page 26
Problem number: Ex 4.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x -a y+y^{2} b -c \,x^{2 a}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 42
dsolve(x*diff(y(x),x)-a*y(x)+b*y(x)^2=c*x^(2*a),y(x), singsol=all)
\[ y \relax (x ) = -\frac {i \tan \left (\frac {i x^{a} \sqrt {c}\, \sqrt {b}-c_{1} a}{a}\right ) \sqrt {c}\, x^{a}}{\sqrt {b}} \]
✓ Solution by Mathematica
Time used: 0.335 (sec). Leaf size: 145
DSolve[x*y'[x]-a*y[x]+b*y[x]^2==c*x^(2*a),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {c} x^a \left (-\cosh \left (\frac {\sqrt {b} \sqrt {c} x^a}{a}\right )+c_1 \sin \left (\frac {\sqrt {-b} \sqrt {c} x^a}{a}\right )\right )}{\sqrt {-b} \left (\sin \left (\frac {\sqrt {-b} \sqrt {c} x^a}{a}\right )+c_1 \cosh \left (\frac {\sqrt {b} \sqrt {c} x^a}{a}\right )\right )} \\ y(x)\to \frac {\sqrt {c} x^a \tanh \left (\frac {\sqrt {b} \sqrt {c} x^a}{a}\right )}{\sqrt {b}} \\ \end{align*}