Internal problem ID [2707]
Internal file name [OUTPUT/2199_Sunday_June_05_2022_02_54_17_AM_90629347/index.tex]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth
edition, 2015
Section: Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page
79
Problem number: Problem 61.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[_Riccati]
\[ \boxed {y^{\prime }+p \left (x \right ) y+q \left (x \right ) y^{2}=r \left (x \right )} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -p \left (x \right ) y -q \left (x \right ) y^{2}+r \left (x \right ) \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -p \left (x \right ) y -q \left (x \right ) y^{2}+r \left (x \right ) \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=r \left (x \right )\), \(f_1(x)=-p \left (x \right )\) and \(f_2(x)=-q \left (x \right )\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-q \left (x \right ) 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' &=-q^{\prime }\left (x \right )\\ f_1 f_2 &=p \left (x \right ) q \left (x \right )\\ f_2^2 f_0 &=r \left (x \right ) q \left (x \right )^{2} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -q \left (x \right ) u^{\prime \prime }\left (x \right )-\left (p \left (x \right ) q \left (x \right )-q^{\prime }\left (x \right )\right ) u^{\prime }\left (x \right )+r \left (x \right ) q \left (x \right )^{2} u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{-q \left (x \right ) r \left (x \right ) \textit {\_Y} \left (x \right )+\frac {\left (p \left (x \right ) q \left (x \right )-q^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{q \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {d}{d x}\operatorname {DESol}\left (\left \{-q \left (x \right ) r \left (x \right ) \textit {\_Y} \left (x \right )+\frac {\left (p \left (x \right ) q \left (x \right )-q^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{q \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = \frac {\frac {d}{d x}\operatorname {DESol}\left (\left \{-q \left (x \right ) r \left (x \right ) \textit {\_Y} \left (x \right )+\frac {\left (p \left (x \right ) q \left (x \right )-q^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{q \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{q \left (x \right ) \operatorname {DESol}\left (\left \{-q \left (x \right ) r \left (x \right ) \textit {\_Y} \left (x \right )+\frac {\left (p \left (x \right ) q \left (x \right )-q^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{q \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\frac {d}{d x}\operatorname {DESol}\left (\left \{-q \left (x \right ) r \left (x \right ) \textit {\_Y} \left (x \right )+\frac {\left (p \left (x \right ) q \left (x \right )-q^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{q \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{q \left (x \right ) \operatorname {DESol}\left (\left \{-q \left (x \right ) r \left (x \right ) \textit {\_Y} \left (x \right )+\frac {\left (p \left (x \right ) q \left (x \right )-q^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{q \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\frac {d}{d x}\operatorname {DESol}\left (\left \{-q \left (x \right ) r \left (x \right ) \textit {\_Y} \left (x \right )+\frac {\left (p \left (x \right ) q \left (x \right )-q^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{q \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{q \left (x \right ) \operatorname {DESol}\left (\left \{-q \left (x \right ) r \left (x \right ) \textit {\_Y} \left (x \right )+\frac {\left (p \left (x \right ) q \left (x \right )-q^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{q \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\frac {d}{d x}\operatorname {DESol}\left (\left \{-q \left (x \right ) r \left (x \right ) \textit {\_Y} \left (x \right )+\frac {\left (p \left (x \right ) q \left (x \right )-q^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{q \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{q \left (x \right ) \operatorname {DESol}\left (\left \{-q \left (x \right ) r \left (x \right ) \textit {\_Y} \left (x \right )+\frac {\left (p \left (x \right ) q \left (x \right )-q^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{q \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+p \left (x \right ) y+q \left (x \right ) y^{2}=r \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-p \left (x \right ) y-q \left (x \right ) y^{2}+r \left (x \right ) \end {array} \]
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) = -(p(x)*q(x)-(diff(q(x), x)))*(diff(y(x), x))/q(x)+r(x)*q(x)*y(x), y(x) Methods for second order ODEs: -> Trying a change of variables to reduce to Bernoulli -> Calling odsolve with the ODE`, diff(y(x), x)-(-q(x)*y(x)^2+y(x)-p(x)*y(x)*x+x^2*r(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`
✗ Solution by Maple
dsolve(diff(y(x),x)+p(x)*y(x)+q(x)*y(x)^2=r(x),y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]+p[x]*y[x]+q[x]*y[x]^2==r[x],y[x],x,IncludeSingularSolutions -> True]
Not solved