Internal problem ID [13300]
Internal file name [OUTPUT/12472_Wednesday_February_14_2024_02_06_33_AM_41829624/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 4. SEPARABLE FIRST ORDER EQUATIONS. Additional exercises. page
90
Problem number: 4.3 (c).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[_rational, _Riccati]
\[ \boxed {y^{\prime } x -\left (x -y\right )^{2}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {\left (-x +y \right )^{2}}{x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = x -2 y +\frac {y^{2}}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=x\), \(f_1(x)=-2\) and \(f_2(x)=\frac {1}{x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {u}{x}} \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' &=-\frac {1}{x^{2}}\\ f_1 f_2 &=-\frac {2}{x}\\ f_2^2 f_0 &=\frac {1}{x} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {u^{\prime \prime }\left (x \right )}{x}-\left (-\frac {2}{x}-\frac {1}{x^{2}}\right ) u^{\prime }\left (x \right )+\frac {u \left (x \right )}{x} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = {\mathrm e}^{-x} \left (\operatorname {BesselI}\left (0, 2 \sqrt {x}\right ) c_{1} +\operatorname {BesselK}\left (0, 2 \sqrt {x}\right ) c_{2} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {{\mathrm e}^{-x} \left (\sqrt {x}\, \operatorname {BesselK}\left (0, 2 \sqrt {x}\right ) c_{2} +\operatorname {BesselK}\left (1, 2 \sqrt {x}\right ) c_{2} +c_{1} \left (\sqrt {x}\, \operatorname {BesselI}\left (0, 2 \sqrt {x}\right )-\operatorname {BesselI}\left (1, 2 \sqrt {x}\right )\right )\right )}{\sqrt {x}} \] Using the above in (1) gives the solution \[ y = \frac {\left (\sqrt {x}\, \operatorname {BesselK}\left (0, 2 \sqrt {x}\right ) c_{2} +\operatorname {BesselK}\left (1, 2 \sqrt {x}\right ) c_{2} +c_{1} \left (\sqrt {x}\, \operatorname {BesselI}\left (0, 2 \sqrt {x}\right )-\operatorname {BesselI}\left (1, 2 \sqrt {x}\right )\right )\right ) \sqrt {x}}{\operatorname {BesselI}\left (0, 2 \sqrt {x}\right ) c_{1} +\operatorname {BesselK}\left (0, 2 \sqrt {x}\right ) c_{2}} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\left (\sqrt {x}\, \operatorname {BesselI}\left (0, 2 \sqrt {x}\right ) c_{3} +\sqrt {x}\, \operatorname {BesselK}\left (0, 2 \sqrt {x}\right )-\operatorname {BesselI}\left (1, 2 \sqrt {x}\right ) c_{3} +\operatorname {BesselK}\left (1, 2 \sqrt {x}\right )\right ) \sqrt {x}}{\operatorname {BesselI}\left (0, 2 \sqrt {x}\right ) c_{3} +\operatorname {BesselK}\left (0, 2 \sqrt {x}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (\sqrt {x}\, \operatorname {BesselI}\left (0, 2 \sqrt {x}\right ) c_{3} +\sqrt {x}\, \operatorname {BesselK}\left (0, 2 \sqrt {x}\right )-\operatorname {BesselI}\left (1, 2 \sqrt {x}\right ) c_{3} +\operatorname {BesselK}\left (1, 2 \sqrt {x}\right )\right ) \sqrt {x}}{\operatorname {BesselI}\left (0, 2 \sqrt {x}\right ) c_{3} +\operatorname {BesselK}\left (0, 2 \sqrt {x}\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (\sqrt {x}\, \operatorname {BesselI}\left (0, 2 \sqrt {x}\right ) c_{3} +\sqrt {x}\, \operatorname {BesselK}\left (0, 2 \sqrt {x}\right )-\operatorname {BesselI}\left (1, 2 \sqrt {x}\right ) c_{3} +\operatorname {BesselK}\left (1, 2 \sqrt {x}\right )\right ) \sqrt {x}}{\operatorname {BesselI}\left (0, 2 \sqrt {x}\right ) c_{3} +\operatorname {BesselK}\left (0, 2 \sqrt {x}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } x -\left (x -y\right )^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (x -y\right )^{2}}{x} \end {array} \]
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 sub-methods: trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -(2*x+1)*(diff(y(x), x))/x-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 <- Bessel successful <- special function solution successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 67
dsolve(x*diff(y(x),x)=(x-y(x))^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sqrt {x}\, \left (\left (\operatorname {BesselK}\left (0, 2 \sqrt {x}\right ) c_{1} +\operatorname {BesselI}\left (0, 2 \sqrt {x}\right )\right ) \sqrt {x}+\operatorname {BesselK}\left (1, 2 \sqrt {x}\right ) c_{1} -\operatorname {BesselI}\left (1, 2 \sqrt {x}\right )\right )}{\operatorname {BesselK}\left (0, 2 \sqrt {x}\right ) c_{1} +\operatorname {BesselI}\left (0, 2 \sqrt {x}\right )} \]
✓ Solution by Mathematica
Time used: 0.251 (sec). Leaf size: 121
DSolve[x*y'[x]==(x-y[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x K_0\left (2 \sqrt {x}\right )+\sqrt {x} K_1\left (2 \sqrt {x}\right )+c_1 x \operatorname {BesselI}\left (0,2 \sqrt {x}\right )-c_1 \sqrt {x} \operatorname {BesselI}\left (1,2 \sqrt {x}\right )}{K_0\left (2 \sqrt {x}\right )+c_1 \operatorname {BesselI}\left (0,2 \sqrt {x}\right )} \\ y(x)\to x-\frac {\sqrt {x} \operatorname {BesselI}\left (1,2 \sqrt {x}\right )}{\operatorname {BesselI}\left (0,2 \sqrt {x}\right )} \\ \end{align*}