2.1.85 problem 84
Internal
problem
ID
[8223]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
1.0
Problem
number
:
84
Date
solved
:
Tuesday, November 12, 2024 at 11:07:36 PM
CAS
classification
:
[_Riccati]
Solve
\begin{align*} y^{\prime }&=x^{2}+y^{2}-1 \end{align*}
Solved as first order ode of type Riccati
Time used: 0.345 (sec)
In canonical form the ODE is
\begin{align*} y' &= F(x,y)\\ &= x^{2}+y^{2}-1 \end{align*}
This is a Riccati ODE. Comparing the ODE to solve
\[ y' = x^{2}+y^{2}-1 \]
With Riccati ODE standard form
\[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]
Shows that \(f_0(x)=x^{2}-1\) , \(f_1(x)=0\) 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^{2}-1 \end{align*}
Substituting the above terms back in equation (2) gives
\begin{align*} u^{\prime \prime }\left (x \right )+\left (x^{2}-1\right ) u \left (x \right ) = 0 \end{align*}
Solution obtained is
\[
u \left (x \right ) = \frac {c_1 \operatorname {WhittakerM}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+c_2 \operatorname {WhittakerW}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )}{\sqrt {x}}
\]
Taking derivative gives
\[
u^{\prime }\left (x \right ) = \frac {2 i c_1 \left (\left (\frac {1}{2}-\frac {1}{4 x^{2}}\right ) \operatorname {WhittakerM}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) \operatorname {WhittakerM}\left (1+\frac {i}{4}, \frac {1}{4}, i x^{2}\right )}{x^{2}}\right ) x +2 i c_2 \left (\left (\frac {1}{2}-\frac {1}{4 x^{2}}\right ) \operatorname {WhittakerW}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+\frac {i \operatorname {WhittakerW}\left (1+\frac {i}{4}, \frac {1}{4}, i x^{2}\right )}{x^{2}}\right ) x}{\sqrt {x}}-\frac {c_1 \operatorname {WhittakerM}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+c_2 \operatorname {WhittakerW}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )}{2 x^{{3}/{2}}}
\]
Doing change of constants, the solution
becomes
\[
y = -\frac {\left (\frac {2 i c_5 \left (\left (\frac {1}{2}-\frac {1}{4 x^{2}}\right ) \operatorname {WhittakerM}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) \operatorname {WhittakerM}\left (1+\frac {i}{4}, \frac {1}{4}, i x^{2}\right )}{x^{2}}\right ) x +2 i \left (\left (\frac {1}{2}-\frac {1}{4 x^{2}}\right ) \operatorname {WhittakerW}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+\frac {i \operatorname {WhittakerW}\left (1+\frac {i}{4}, \frac {1}{4}, i x^{2}\right )}{x^{2}}\right ) x}{\sqrt {x}}-\frac {c_5 \operatorname {WhittakerM}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+\operatorname {WhittakerW}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )}{2 x^{{3}/{2}}}\right ) \sqrt {x}}{c_5 \operatorname {WhittakerM}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+\operatorname {WhittakerW}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )}
\]
Figure 2.173: Slope field plot
\(y^{\prime } = x^{2}+y^{2}-1\)
Summary of solutions found
\begin{align*}
y &= \frac {\left (-3-i\right ) c_5 \operatorname {WhittakerM}\left (1+\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+4 \operatorname {WhittakerW}\left (1+\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+\left (-2 i x^{2}+i+1\right ) c_5 \operatorname {WhittakerM}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+\left (-2 i x^{2}+i+1\right ) \operatorname {WhittakerW}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )}{2 x \left (c_5 \operatorname {WhittakerM}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+\operatorname {WhittakerW}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )\right )} \\
\end{align*}
Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=x^{2}+y \left (x \right )^{2}-1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=x^{2}+y \left (x \right )^{2}-1 \end {array} \]
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
trying Riccati sub-methods:
trying Riccati to 2nd Order
-> Calling odsolve with the ODE ` , diff(diff(y(x), x), x) = (-x^2+1)*y(x), y(x) ` *** Sublevel 2 ***
Methods for second order ODEs:
--- Trying classification methods ---
trying a quadrature
checking if the LODE has constant coefficients
checking if the LODE is of Euler type
trying a symmetry of the form [xi=0, eta=F(x)]
checking if the LODE is missing y
-> Trying a Liouvillian solution using Kovacics algorithm
<- No Liouvillian solutions exists
-> Trying a solution in terms of special functions:
-> Bessel
-> elliptic
-> Legendre
-> Whittaker
-> hyper3: Equivalence to 1F1 under a power @ Moebius
<- hyper3 successful: received ODE is equivalent to the 1F1 ODE
<- Whittaker successful
<- special function solution successful
<- Riccati to 2nd Order successful `
Maple dsolve solution
Solving time : 0.003
(sec)
Leaf size : 85
dsolve ( diff ( y ( x ), x ) = x^2+y(x)^2-1,
y(x),singsol=all)
\[
y = \frac {\left (-3-i\right ) \operatorname {WhittakerM}\left (1+\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+4 \operatorname {WhittakerW}\left (1+\frac {i}{4}, \frac {1}{4}, i x^{2}\right ) c_{1} +\left (-2 i x^{2}+i+1\right ) \operatorname {WhittakerM}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+\left (-2 i x^{2}+i+1\right ) c_{1} \operatorname {WhittakerW}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )}{2 x \left (c_{1} \operatorname {WhittakerW}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )+\operatorname {WhittakerM}\left (\frac {i}{4}, \frac {1}{4}, i x^{2}\right )\right )}
\]
Mathematica DSolve solution
Solving time : 0.225
(sec)
Leaf size : 153
DSolve [{ D [ y [ x ], x ]== x ^2+ y [ x ]^2-1,{}},
y[x],x,IncludeSingularSolutions-> True ]
\begin{align*}
y(x)\to \frac {i \left (x \operatorname {ParabolicCylinderD}\left (-\frac {1}{2}-\frac {i}{2},(-1+i) x\right )+(1+i) \operatorname {ParabolicCylinderD}\left (\frac {1}{2}-\frac {i}{2},(-1+i) x\right )-c_1 x \operatorname {ParabolicCylinderD}\left (-\frac {1}{2}+\frac {i}{2},(1+i) x\right )+(1-i) c_1 \operatorname {ParabolicCylinderD}\left (\frac {1}{2}+\frac {i}{2},(1+i) x\right )\right )}{\operatorname {ParabolicCylinderD}\left (-\frac {1}{2}-\frac {i}{2},(-1+i) x\right )+c_1 \operatorname {ParabolicCylinderD}\left (-\frac {1}{2}+\frac {i}{2},(1+i) x\right )} \\
y(x)\to \frac {(1+i) \operatorname {ParabolicCylinderD}\left (\frac {1}{2}+\frac {i}{2},(1+i) x\right )}{\operatorname {ParabolicCylinderD}\left (-\frac {1}{2}+\frac {i}{2},(1+i) x\right )}-i x \\
\end{align*}