1.12 problem Example 3.12

Internal problem ID [5845]
Internal file name [OUTPUT/5093_Sunday_June_05_2022_03_24_10_PM_796261/index.tex]

Book: THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section: Chapter 3. Ordinary Differential Equations. Section 3.2 FIRST ORDER ODE. Page 114
Problem number: Example 3.12.
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]

\[ \boxed {y^{\prime }+\frac {y}{t}+y^{2}=-1} \]

1.12.1 Solving as riccati ode

In canonical form the ODE is \begin {align*} y' &= F(t,y)\\ &= -\frac {y^{2} t +t +y}{t} \end {align*}

This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {y}{t}-1-y^{2} \] With Riccati ODE standard form \[ y' = f_0(t)+ f_1(t)y+f_2(t)y^{2} \] Shows that \(f_0(t)=-1\), \(f_1(t)=-\frac {1}{t}\) and \(f_2(t)=-1\). 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''(t) -\left ( f_2' + f_1 f_2 \right ) u'(t) + f_2^2 f_0 u(t) &= 0 \tag {2} \end {align*}

But \begin {align*} f_2' &=0\\ f_1 f_2 &=\frac {1}{t}\\ f_2^2 f_0 &=-1 \end {align*}

Substituting the above terms back in equation (2) gives \begin {align*} -u^{\prime \prime }\left (t \right )-\frac {u^{\prime }\left (t \right )}{t}-u \left (t \right ) &=0 \end {align*}

Solving the above ODE (this ode solved using Maple, not this program), gives

\[ u \left (t \right ) = c_{1} \operatorname {BesselJ}\left (0, t\right )+c_{2} \operatorname {BesselY}\left (0, t\right ) \] The above shows that \[ u^{\prime }\left (t \right ) = -c_{1} \operatorname {BesselJ}\left (1, t\right )-c_{2} \operatorname {BesselY}\left (1, t\right ) \] Using the above in (1) gives the solution \[ y = \frac {-c_{1} \operatorname {BesselJ}\left (1, t\right )-c_{2} \operatorname {BesselY}\left (1, t\right )}{c_{1} \operatorname {BesselJ}\left (0, t\right )+c_{2} \operatorname {BesselY}\left (0, t\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution

\[ y = \frac {-c_{3} \operatorname {BesselJ}\left (1, t\right )-\operatorname {BesselY}\left (1, t\right )}{c_{3} \operatorname {BesselJ}\left (0, t\right )+\operatorname {BesselY}\left (0, t\right )} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-c_{3} \operatorname {BesselJ}\left (1, t\right )-\operatorname {BesselY}\left (1, t\right )}{c_{3} \operatorname {BesselJ}\left (0, t\right )+\operatorname {BesselY}\left (0, t\right )} \\ \end{align*}

Figure 30: Slope field plot

Verification of solutions

\[ y = \frac {-c_{3} \operatorname {BesselJ}\left (1, t\right )-\operatorname {BesselY}\left (1, t\right )}{c_{3} \operatorname {BesselJ}\left (0, t\right )+\operatorname {BesselY}\left (0, t\right )} \] Verified OK.

1.12.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{2} t +t y^{\prime }+y+t =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {y^{2} t +t +y}{t} \end {array} \]

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 a Liouvillian solution using Kovacics algorithm 
      <- No Liouvillian solutions exists 
        -> searching for a solution in terms of Whittaker functions 
        <- solution in terms of Whittaker functions successful 
   <- Abel AIR successful: ODE belongs to the 1F1 2-parameter class`
 

✓ Solution by Maple

Time used: 0.063 (sec). Leaf size: 35

dsolve(diff(y(t),t)=-y(t)/t-1-y(t)^2,y(t), singsol=all)
 

\[ y \left (t \right ) = \frac {-i \operatorname {BesselK}\left (1, i t \right ) c_{1} -\operatorname {BesselJ}\left (1, t\right )}{\operatorname {BesselK}\left (0, i t \right ) c_{1} +\operatorname {BesselJ}\left (0, t\right )} \]

✓ Solution by Mathematica

Time used: 0.189 (sec). Leaf size: 43

DSolve[y'[t]==-y[t]/t-1-y[t]^2,y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to -\frac {\operatorname {BesselY}(1,t)+c_1 \operatorname {BesselJ}(1,t)}{\operatorname {BesselY}(0,t)+c_1 \operatorname {BesselJ}(0,t)} \\ y(t)\to -\frac {\operatorname {BesselJ}(1,t)}{\operatorname {BesselJ}(0,t)} \\ \end{align*}