2.1.11 problem 11

Internal problem ID [8671]
Book : First order enumerated odes
Section : section 1
Problem number : 11
Date solved : Tuesday, December 17, 2024 at 12:57:23 PM
CAS classification : [[_Riccati, _special]]

Solve

\begin{align*} y^{\prime }&=a x +b y^{2} \end{align*}

Solved as first order ode of type reduced Riccati

Time used: 0.135 (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 &= a\\ b &= b\\ 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 {a b}\, x^{{3}/{2}}}{3}\right )+c_2 \operatorname {BesselY}\left (\frac {1}{3}, \frac {2 \sqrt {a b}\, 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

\[ y = \frac {\left (-\operatorname {BesselY}\left (-\frac {2}{3}, \frac {2 \sqrt {a b}\, x^{{3}/{2}}}{3}\right ) c_2 -\operatorname {BesselJ}\left (-\frac {2}{3}, \frac {2 \sqrt {a b}\, x^{{3}/{2}}}{3}\right ) c_1 \right ) \sqrt {a b}\, \sqrt {x}}{b \left (c_1 \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 \sqrt {a b}\, x^{{3}/{2}}}{3}\right )+c_2 \operatorname {BesselY}\left (\frac {1}{3}, \frac {2 \sqrt {a b}\, x^{{3}/{2}}}{3}\right )\right )} \]

Letting \(c_2 = 1\) the above becomes

\[ y = \frac {\left (-\operatorname {BesselY}\left (-\frac {2}{3}, \frac {2 \sqrt {a b}\, x^{{3}/{2}}}{3}\right )-\operatorname {BesselJ}\left (-\frac {2}{3}, \frac {2 \sqrt {a b}\, x^{{3}/{2}}}{3}\right ) c_1 \right ) \sqrt {a b}\, \sqrt {x}}{b \left (c_1 \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 \sqrt {a b}\, x^{{3}/{2}}}{3}\right )+\operatorname {BesselY}\left (\frac {1}{3}, \frac {2 \sqrt {a b}\, x^{{3}/{2}}}{3}\right )\right )} \]

Summary of solutions found

\begin{align*} y &= \frac {\left (-\operatorname {BesselY}\left (-\frac {2}{3}, \frac {2 \sqrt {a b}\, x^{{3}/{2}}}{3}\right )-\operatorname {BesselJ}\left (-\frac {2}{3}, \frac {2 \sqrt {a b}\, x^{{3}/{2}}}{3}\right ) c_1 \right ) \sqrt {a b}\, \sqrt {x}}{b \left (c_1 \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 \sqrt {a b}\, x^{{3}/{2}}}{3}\right )+\operatorname {BesselY}\left (\frac {1}{3}, \frac {2 \sqrt {a b}\, x^{{3}/{2}}}{3}\right )\right )} \\ \end{align*}

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=x a +b y \left (x \right )^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=x a +b y \left (x \right )^{2} \end {array} \]

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 : 0.003 (sec)
Leaf size : 59

dsolve(diff(y(x),x) = a*x+b*y(x)^2, 
       y(x),singsol=all)
 

\[ y = \frac {\left (b a \right )^{{1}/{3}} \left (\operatorname {AiryAi}\left (1, -\left (b a \right )^{{1}/{3}} x \right ) c_{1} +\operatorname {AiryBi}\left (1, -\left (b a \right )^{{1}/{3}} x \right )\right )}{b \left (c_{1} \operatorname {AiryAi}\left (-\left (b a \right )^{{1}/{3}} x \right )+\operatorname {AiryBi}\left (-\left (b a \right )^{{1}/{3}} x \right )\right )} \]

Mathematica DSolve solution

Solving time : 0.156 (sec)
Leaf size : 331

DSolve[{D[y[x],x]==a*x+b*y[x]^2,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {\sqrt {a} \sqrt {b} x^{3/2} \left (-2 \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )+c_1 \left (\operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )-\operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )\right )\right )-c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )}{2 b x \left (\operatorname {BesselJ}\left (\frac {1}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )\right )} \\ y(x)\to -\frac {\sqrt {a} \sqrt {b} x^{3/2} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )-\sqrt {a} \sqrt {b} x^{3/2} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )}{2 b x \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )} \\ \end{align*}