2.1.81 problem 80

Internal problem ID [8219]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 80
Date solved : Tuesday, November 12, 2024 at 11:07:35 PM
CAS classification : [[_Riccati, _special]]

Solve

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

Solved as first order ode of type reduced Riccati

Time used: 0.086 (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 &= 1\\ b &= 1\\ n &= 2 \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*}

Since \(ab>0\) then EQ(1) gives

\begin{align*} k &= 2\\ w &= \sqrt {x}\, \left (c_1 \operatorname {BesselJ}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+c_2 \operatorname {BesselY}\left (\frac {1}{4}, \frac {x^{2}}{2}\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

\[ y = \frac {\left (-\operatorname {BesselY}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) c_2 -\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) c_1 \right ) x}{c_1 \operatorname {BesselJ}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+c_2 \operatorname {BesselY}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )} \]

Letting \(c_2 = 1\) the above becomes

\[ y = \frac {\left (-\operatorname {BesselY}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right )-\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) c_1 \right ) x}{c_1 \operatorname {BesselJ}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselY}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )} \]

Figure 2.164: Slope field plot
\(y^{\prime } = x^{2}+y^{2}\)

Summary of solutions found

\begin{align*} y &= \frac {\left (-\operatorname {BesselY}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right )-\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) c_1 \right ) x}{c_1 \operatorname {BesselJ}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselY}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )} \\ \end{align*}

Maple step by step solution

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

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 : 0.013 (sec)
Leaf size : 43

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

\[ y = -\frac {x \left (\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) c_{1} +\operatorname {BesselY}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right )\right )}{c_{1} \operatorname {BesselJ}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselY}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )} \]

Mathematica DSolve solution

Solving time : 0.134 (sec)
Leaf size : 169

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

\begin{align*} y(x)\to \frac {x^2 \left (-2 \operatorname {BesselJ}\left (-\frac {3}{4},\frac {x^2}{2}\right )+c_1 \left (\operatorname {BesselJ}\left (\frac {3}{4},\frac {x^2}{2}\right )-\operatorname {BesselJ}\left (-\frac {5}{4},\frac {x^2}{2}\right )\right )\right )-c_1 \operatorname {BesselJ}\left (-\frac {1}{4},\frac {x^2}{2}\right )}{2 x \left (\operatorname {BesselJ}\left (\frac {1}{4},\frac {x^2}{2}\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{4},\frac {x^2}{2}\right )\right )} \\ y(x)\to -\frac {x^2 \operatorname {BesselJ}\left (-\frac {5}{4},\frac {x^2}{2}\right )-x^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {x^2}{2}\right )+\operatorname {BesselJ}\left (-\frac {1}{4},\frac {x^2}{2}\right )}{2 x \operatorname {BesselJ}\left (-\frac {1}{4},\frac {x^2}{2}\right )} \\ \end{align*}