2.41 problem 41
Internal
problem
ID
[8130]
Book
:
Second
order
enumerated
odes
Section
:
section
2
Problem
number
:
41
Date
solved
:
Monday, October 21, 2024 at 04:54:08 PM
CAS
classification
:
[[_Riccati, _special]]
Solve
\begin{align*} y^{\prime }&=x -y^{2} \end{align*}
2.41.1 Solved as first order ode of type Riccati
Time used: 0.266 (sec)
In canonical form the ODE is
\begin{align*} y' &= F(x,y)\\ &= -y^{2}+x \end{align*}
This is a Riccati ODE. Comparing the ODE to solve
\[ y' = -y^{2}+x \]
With Riccati ODE standard form
\[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]
Shows that \(f_0(x)=x\) , \(f_1(x)=0\) and \(f_2(x)=-1\) . Let
\begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-u} \tag {1} \end{align*}
Using the above substitution in the given ODE results (after some simplification)in a second
order ODE to solve for \(u(x)\) which is
\begin{align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end{align*}
But
\begin{align*} f_2' &=0\\ f_1 f_2 &=0\\ f_2^2 f_0 &=x \end{align*}
Substituting the above terms back in equation (2) gives
\begin{align*} -u^{\prime \prime }\left (x \right )+x u \left (x \right ) = 0 \end{align*}
This is Airy ODE. It has the general form
\[ a \frac {d^{2}u}{d x^{2}} + b \frac {d u}{d x} + c x u = F(x) \]
Where in this case
\begin{align*} a &= -1\\ b &= 0\\ c &= 1\\ F &= 0 \end{align*}
Therefore the solution to the homogeneous Airy ODE becomes
\[
u = c_1 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+c_2 \operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )
\]
Will add steps showing
solving for IC soon.
Taking derivative gives
\[
u^{\prime }\left (x \right ) = -c_1 \left (-1\right )^{{1}/{3}} \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right )-c_2 \left (-1\right )^{{1}/{3}} \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right )
\]
Doing change of constants, the solution becomes
\[
y = \frac {-c_3 \left (-1\right )^{{1}/{3}} \operatorname {AiryAi}\left (1, -x \left (-1\right )^{{1}/{3}}\right )-\left (-1\right )^{{1}/{3}} \operatorname {AiryBi}\left (1, -x \left (-1\right )^{{1}/{3}}\right )}{c_3 \operatorname {AiryAi}\left (-x \left (-1\right )^{{1}/{3}}\right )+\operatorname {AiryBi}\left (-x \left (-1\right )^{{1}/{3}}\right )}
\]
Figure 163: Slope field plot
\(y^{\prime } = x -y^{2}\)
2.41.2 Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=x -y^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=x -y^{2} \end {array} \]
2.41.3 Maple trace
Methods for first order ODEs:
2.41.4 Maple dsolve solution
Solving time : 0.003
(sec)
Leaf size : 23
dsolve ( diff ( y ( x ), x ) = x-y(x)^2,
y(x),singsol=all)
\[
y = \frac {c_1 \operatorname {AiryAi}\left (1, x\right )+\operatorname {AiryBi}\left (1, x\right )}{c_1 \operatorname {AiryAi}\left (x \right )+\operatorname {AiryBi}\left (x \right )}
\]
2.41.5 Mathematica DSolve solution
Solving time : 0.18
(sec)
Leaf size : 223
DSolve [{ D [ y [ x ], x ]== x - y [ x ]^2,{}},
y[x],x,IncludeSingularSolutions-> True ]
\begin{align*}
y(x)\to -\frac {-i x^{3/2} \left (2 \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2}{3} i x^{3/2}\right )+c_1 \left (\operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} i x^{3/2}\right )-\operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} i x^{3/2}\right )\right )\right )-c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i x^{3/2}\right )}{2 x \left (\operatorname {BesselJ}\left (\frac {1}{3},\frac {2}{3} i x^{3/2}\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i x^{3/2}\right )\right )} \\
y(x)\to \frac {i x^{3/2} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} i x^{3/2}\right )-i x^{3/2} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} i x^{3/2}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i x^{3/2}\right )}{2 x \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i x^{3/2}\right )} \\
\end{align*}