problem 60

Internal problem ID [3324]
Internal ﬁle name [OUTPUT/2816_Sunday_June_05_2022_08_41_03_AM_67836189/index.tex]

Book: Ordinary diﬀerential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 3
Problem number: 60.
ODE order: 1.
ODE degree: 1.

3.6.1 Solving as riccati ode

In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f \left (x \right )+a y +b \,y^{2} \end {align*}

This is a Riccati ODE. Comparing the ODE to solve \[ y' = f \left (x \right )+a y +b \,y^{2} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=f \left (x \right )\), \(f_1(x)=a\) and \(f_2(x)=b\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{b 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 &=a b\\ f_2^2 f_0 &=b^{2} f \left (x \right ) \end {align*}

Substituting the above terms back in equation (2) gives \begin {align*} b u^{\prime \prime }\left (x \right )-a b u^{\prime }\left (x \right )+b^{2} f \left (x \right ) u \left (x \right ) &=0 \end {align*}

\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{f \left (x \right ) b \textit {\_Y} \left (x \right )-a \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {d}{d x}\operatorname {DESol}\left (\left \{f \left (x \right ) b \textit {\_Y} \left (x \right )-a \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\frac {d}{d x}\operatorname {DESol}\left (\left \{f \left (x \right ) b \textit {\_Y} \left (x \right )-a \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{b \operatorname {DESol}\left (\left \{f \left (x \right ) b \textit {\_Y} \left (x \right )-a \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution

\[ y = -\frac {\frac {d}{d x}\operatorname {DESol}\left (\left \{f \left (x \right ) b \textit {\_Y} \left (x \right )-a \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{b \operatorname {DESol}\left (\left \{f \left (x \right ) b \textit {\_Y} \left (x \right )-a \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\frac {d}{d x}\operatorname {DESol}\left (\left \{f \left (x \right ) b \textit {\_Y} \left (x \right )-a \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{b \operatorname {DESol}\left (\left \{f \left (x \right ) b \textit {\_Y} \left (x \right )-a \textit {\_Y}^{\prime }\left (x \right )+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}

Veriﬁcation of solutions

3.6.2 Maple step by step solution

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

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 sub-methods: 
   trying Riccati_symmetries 
   trying Riccati to 2nd Order 
   -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = a*(diff(y(x), x))-b*f(x)*y(x), y(x)`      *** Sublevel 2 *** 
      Methods for second order ODEs: 
   -> Trying a change of variables to reduce to Bernoulli 
   -> Calling odsolve with the ODE`, diff(y(x), x)-(b*y(x)^2+y(x)+a*y(x)*x+x^2*f(x))/x, y(x), explicit`      *** Sublevel 2 *** 
      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_symmetries 
      trying inverse_Riccati 
      trying 1st order ODE linearizable_by_differentiation 
   -> trying a symmetry pattern of the form [F(x)*G(y), 0] 
   -> trying a symmetry pattern of the form [0, F(x)*G(y)] 
   -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
trying inverse_Riccati 
trying 1st order ODE linearizable_by_differentiation 
--- Trying Lie symmetry methods, 1st order --- 
`, `-> Computing symmetries using: way = 4 
`, `-> Computing symmetries using: way = 2 
`, `-> Computing symmetries using: way = 6`

3.6 problem 60

3.6.1 Solving as riccati ode

3.6.2 Maple step by step solution