2.1.22 Problem 22

Internal problem ID [9006]
Book : First order enumerated odes
Section : section 1
Problem number : 22
Date solved : Sunday, March 30, 2025 at 01:58:46 PM
CAS classification : [_rational, _Riccati]

Unknown ode type.

✓ Maple. Time used: 0.009 (sec). Leaf size: 106

ode:=c*diff(y(x),x) = (a*x+b*y(x)^2)/r/x^2; 
dsolve(ode,y(x), singsol=all);
 

$$y=\frac{a(\mathrm{BesselY}(0,2\sqrt{\frac{ab}{x{r}^2{c}^2}})c_1+\mathrm{BesselJ}(0,2\sqrt{\frac{ab}{x{r}^2{c}^2}}))}{cr\sqrt{\frac{ab}{x{r}^2{c}^2}}(c_1\mathrm{BesselY}(1,2\sqrt{\frac{ab}{x{r}^2{c}^2}})+\mathrm{BesselJ}(1,2\sqrt{\frac{ab}{x{r}^2{c}^2}}))}$$

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying Chini 
differential order: 1; looking for linear symmetries 
trying exact 
Looking for potential symmetries 
trying Riccati 
trying Riccati sub-methods: 
      -> Trying a Liouvillian solution using Kovacics algorithm 
      <- No Liouvillian solutions exists 
   <- Abel AIR successful: ODE belongs to the 0F1 1-parameter (Bessel type) cla\ 
ss
 

Maple step by step

$$\begin{array}{lll}& & \text{Let’s solve}\\ & & c\left(\frac{d}{dx}y\left(x\right)\right)=\frac{ax+by{\left(x\right)}^2}{r{x}^2}\\ \bullet & & \text{Highest derivative means the order of the ODE is}1\\ & & \frac{d}{dx}y\left(x\right)\\ \bullet & & \text{Solve for the highest derivative}\\ & & \frac{d}{dx}y\left(x\right)=\frac{ax+by{\left(x\right)}^2}{r{x}^{2}c}\end{array}$$

✓ Mathematica. Time used: 0.359 (sec). Leaf size: 492

ode=c*D[y[x],x]==(a*x+b*y[x]^2)/(r*x^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

$$\begin{array}{r}y(x)\to \frac{2\sqrt{a}\sqrt{b}\mathrm{BesselY}(0,\frac{2\sqrt{a}\sqrt{b}\sqrt{\frac{1}{x}}}{cr})+\frac{2cr\mathrm{BesselY}(1,\frac{2\sqrt{a}\sqrt{b}\sqrt{\frac{1}{x}}}{cr})}{\sqrt{\frac{1}{x}}}-2\sqrt{a}\sqrt{b}\mathrm{BesselY}(2,\frac{2\sqrt{a}\sqrt{b}\sqrt{\frac{1}{x}}}{cr})-i\sqrt{a}\sqrt{b}{c}_1\mathrm{BesselJ}(0,\frac{2\sqrt{a}\sqrt{b}\sqrt{\frac{1}{x}}}{cr})-\frac{ic{c}_{1}r\mathrm{BesselJ}(1,\frac{2\sqrt{a}\sqrt{b}\sqrt{\frac{1}{x}}}{cr})}{\sqrt{\frac{1}{x}}}+i\sqrt{a}\sqrt{b}{c}_1\mathrm{BesselJ}(2,\frac{2\sqrt{a}\sqrt{b}\sqrt{\frac{1}{x}}}{cr})}{2b\sqrt{\frac{1}{x}}(2\mathrm{BesselY}(1,\frac{2\sqrt{a}\sqrt{b}\sqrt{\frac{1}{x}}}{cr})-i{c}_1\mathrm{BesselJ}(1,\frac{2\sqrt{a}\sqrt{b}\sqrt{\frac{1}{x}}}{cr}))}\\ y(x)\to \frac{x(\sqrt{a}\sqrt{b}\sqrt{\frac{1}{x}}\mathrm{BesselJ}(0,\frac{2\sqrt{a}\sqrt{b}\sqrt{\frac{1}{x}}}{cr})+cr\mathrm{BesselJ}(1,\frac{2\sqrt{a}\sqrt{b}\sqrt{\frac{1}{x}}}{cr})-\sqrt{a}\sqrt{b}\sqrt{\frac{1}{x}}\mathrm{BesselJ}(2,\frac{2\sqrt{a}\sqrt{b}\sqrt{\frac{1}{x}}}{cr}))}{2b\mathrm{BesselJ}(1,\frac{2\sqrt{a}\sqrt{b}\sqrt{\frac{1}{x}}}{cr})}\end{array}$$

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
r = symbols("r") 
y = Function("y") 
ode = Eq(c*Derivative(y(x), x) - (a*x + b*y(x)**2)/(r*x**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

RecursionError : maximum recursion depth exceeded