60.1.107 problem 107

Internal problem ID [10121]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 107
Date solved : Monday, January 27, 2025 at 06:28:23 PM
CAS classification : [_rational, _Riccati]

\begin{align*} x y^{\prime }+a \,x^{\alpha } y^{2}+b y-c \,x^{\beta }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 240

dsolve(x*diff(y(x),x) + a*x^alpha*y(x)^2 + b*y(x) - c*x^beta=0,y(x), singsol=all)
 

\[ y = \frac {c \,x^{\beta } \left (\operatorname {BesselY}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right )\right )}{-x^{\frac {\alpha }{2}+\frac {\beta }{2}} \left (\operatorname {BesselY}\left (\frac {b +2 \beta +\alpha }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b +2 \beta +\alpha }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right )\right ) \sqrt {-a c}+\left (b +\beta \right ) \left (\operatorname {BesselY}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b +\beta }{\alpha +\beta }, \frac {2 \sqrt {-a c}\, x^{\frac {\alpha }{2}+\frac {\beta }{2}}}{\alpha +\beta }\right )\right )} \]

✓ Solution by Mathematica

Time used: 0.836 (sec). Leaf size: 1474

DSolve[x*D[y[x],x] + a*x^\[Alpha]*y[x]^2 + b*y[x] - c*x^\[Beta]==0,y[x],x,IncludeSingularSolutions -> True]
 

Too large to display