2.1.20 problem 20
Internal
problem
ID
[8680]
Book
:
First
order
enumerated
odes
Section
:
section
1
Problem
number
:
20
Date
solved
:
Tuesday, December 17, 2024 at 12:57:32 PM
CAS
classification
:
[[_Riccati, _special]]
Solve
\begin{align*} c y^{\prime }&=\frac {a x +b y^{2}}{r} \end{align*}
Solved as first order ode of type reduced Riccati
Time used: 0.155 (sec)
This is reduced Riccati ode of the form
\begin{align*} y^{\prime }&=a \,x^{n}+b y^{2} \end{align*}
Comparing the given ode to the above shows that
\begin{align*} a &= \frac {a}{r c}\\ b &= \frac {b}{c r}\\ n &= 1 \end{align*}
Since \(n\neq -2\) then the solution of the reduced Riccati ode is given by
\begin{align*} w & =\sqrt {x}\left \{ \begin {array}[c]{cc} c_{1}\operatorname {BesselJ}\left ( \frac {1}{2k},\frac {1}{k}\sqrt {ab} x^{k}\right ) +c_{2}\operatorname {BesselY}\left ( \frac {1}{2k},\frac {1}{k}\sqrt {ab}x^{k}\right ) & ab>0\\ c_{1}\operatorname {BesselI}\left ( \frac {1}{2k},\frac {1}{k}\sqrt {-ab}x^{k}\right ) +c_{2}\operatorname {BesselK}\left ( \frac {1}{2k},\frac {1}{k}\sqrt {-ab}x^{k}\right ) & ab<0 \end {array} \right . \tag {1}\\ y & =-\frac {1}{b}\frac {w^{\prime }}{w}\nonumber \\ k &=1+\frac {n}{2}\nonumber \end{align*}
EQ(1) gives
\begin{align*} k &= {\frac {3}{2}}\\ w &= \sqrt {x}\, \left (c_1 \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 \sqrt {\frac {a b}{r^{2} c^{2}}}\, x^{{3}/{2}}}{3}\right )+c_2 \operatorname {BesselY}\left (\frac {1}{3}, \frac {2 \sqrt {\frac {a b}{r^{2} c^{2}}}\, x^{{3}/{2}}}{3}\right )\right ) \end{align*}
Therefore the solution becomes
\begin{align*} y & =-\frac {1}{b}\frac {w^{\prime }}{w} \end{align*}
Substituting the value of \(b,w\) found above and simplyfing gives
\[
y = \frac {\left (-\operatorname {BesselY}\left (-\frac {2}{3}, \frac {2 \sqrt {\frac {a b}{r^{2} c^{2}}}\, x^{{3}/{2}}}{3}\right ) c_2 -\operatorname {BesselJ}\left (-\frac {2}{3}, \frac {2 \sqrt {\frac {a b}{r^{2} c^{2}}}\, x^{{3}/{2}}}{3}\right ) c_1 \right ) c r \sqrt {\frac {a b}{r^{2} c^{2}}}\, \sqrt {x}}{b \left (c_1 \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 \sqrt {\frac {a b}{r^{2} c^{2}}}\, x^{{3}/{2}}}{3}\right )+c_2 \operatorname {BesselY}\left (\frac {1}{3}, \frac {2 \sqrt {\frac {a b}{r^{2} c^{2}}}\, x^{{3}/{2}}}{3}\right )\right )}
\]
Letting \(c_2 = 1\) the above becomes
\[
y = \frac {\left (-\operatorname {BesselY}\left (-\frac {2}{3}, \frac {2 \sqrt {\frac {a b}{r^{2} c^{2}}}\, x^{{3}/{2}}}{3}\right )-\operatorname {BesselJ}\left (-\frac {2}{3}, \frac {2 \sqrt {\frac {a b}{r^{2} c^{2}}}\, x^{{3}/{2}}}{3}\right ) c_1 \right ) c r \sqrt {\frac {a b}{r^{2} c^{2}}}\, \sqrt {x}}{b \left (c_1 \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 \sqrt {\frac {a b}{r^{2} c^{2}}}\, x^{{3}/{2}}}{3}\right )+\operatorname {BesselY}\left (\frac {1}{3}, \frac {2 \sqrt {\frac {a b}{r^{2} c^{2}}}\, x^{{3}/{2}}}{3}\right )\right )}
\]
Summary of solutions found
\begin{align*}
y &= \frac {\left (-\operatorname {BesselY}\left (-\frac {2}{3}, \frac {2 \sqrt {\frac {a b}{r^{2} c^{2}}}\, x^{{3}/{2}}}{3}\right )-\operatorname {BesselJ}\left (-\frac {2}{3}, \frac {2 \sqrt {\frac {a b}{r^{2} c^{2}}}\, x^{{3}/{2}}}{3}\right ) c_1 \right ) c r \sqrt {\frac {a b}{r^{2} c^{2}}}\, \sqrt {x}}{b \left (c_1 \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 \sqrt {\frac {a b}{r^{2} c^{2}}}\, x^{{3}/{2}}}{3}\right )+\operatorname {BesselY}\left (\frac {1}{3}, \frac {2 \sqrt {\frac {a b}{r^{2} c^{2}}}\, x^{{3}/{2}}}{3}\right )\right )} \\
\end{align*}
Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & c \left (\frac {d}{d x}y \left (x \right )\right )=\frac {x a +b y \left (x \right )^{2}}{r} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {x a +b y \left (x \right )^{2}}{r c} \end {array} \]
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 Special
<- Riccati Special successful`
Maple dsolve solution
Solving time : 0.004 (sec)
Leaf size : 91
dsolve(c*diff(y(x),x) = (a*x+b*y(x)^2)/r,
y(x),singsol=all)
\[
y = \frac {\left (\frac {b a}{r^{2} c^{2}}\right )^{{1}/{3}} \left (\operatorname {AiryAi}\left (1, -\left (\frac {b a}{r^{2} c^{2}}\right )^{{1}/{3}} x \right ) c_{1} +\operatorname {AiryBi}\left (1, -\left (\frac {b a}{r^{2} c^{2}}\right )^{{1}/{3}} x \right )\right ) r c}{b \left (c_{1} \operatorname {AiryAi}\left (-\left (\frac {b a}{r^{2} c^{2}}\right )^{{1}/{3}} x \right )+\operatorname {AiryBi}\left (-\left (\frac {b a}{r^{2} c^{2}}\right )^{{1}/{3}} x \right )\right )}
\]
Mathematica DSolve solution
Solving time : 0.21 (sec)
Leaf size : 517
DSolve[{c*D[y[x],x]==(a*x+b*y[x]^2)/r,{}},
y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {c r \left (x^{3/2} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} \left (-2 \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )+c_1 \left (\operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )-\operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )\right )\right )-c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )\right )}{2 b x \left (\operatorname {BesselJ}\left (\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )\right )} \\
y(x)\to -\frac {c r \left (x^{3/2} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )-x^{3/2} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )\right )}{2 b x \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {\frac {a}{c r}} \sqrt {\frac {b}{c r}} x^{3/2}\right )} \\
\end{align*}