Internal problem ID [3339]
Internal file name [OUTPUT/2831_Sunday_June_05_2022_08_41_19_AM_55602258/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 3
Problem number: 75.
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]
\[ \boxed {y^{\prime }-g \left (x \right ) y-h \left (x \right ) y^{2}=f \left (x \right )} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f \left (x \right )+g \left (x \right ) y +h \left (x \right ) y^{2} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = f \left (x \right )+g \left (x \right ) y +h \left (x \right ) y^{2} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=f \left (x \right )\), \(f_1(x)=g \left (x \right )\) and \(f_2(x)=h \left (x \right )\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{h \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' &=h^{\prime }\left (x \right )\\ f_1 f_2 &=g \left (x \right ) h \left (x \right )\\ f_2^2 f_0 &=f \left (x \right ) h \left (x \right )^{2} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} h \left (x \right ) u^{\prime \prime }\left (x \right )-\left (g \left (x \right ) h \left (x \right )+h^{\prime }\left (x \right )\right ) u^{\prime }\left (x \right )+f \left (x \right ) h \left (x \right )^{2} u \left (x \right ) &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{f \left (x \right ) h \left (x \right ) \textit {\_Y} \left (x \right )-\frac {\left (g \left (x \right ) h \left (x \right )+h^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{h \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {d}{d x}\operatorname {DESol}\left (\left \{f \left (x \right ) h \left (x \right ) \textit {\_Y} \left (x \right )-\frac {\left (g \left (x \right ) h \left (x \right )+h^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{h \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\frac {d}{d x}\operatorname {DESol}\left (\left \{f \left (x \right ) h \left (x \right ) \textit {\_Y} \left (x \right )-\frac {\left (g \left (x \right ) h \left (x \right )+h^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{h \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{h \left (x \right ) \operatorname {DESol}\left (\left \{f \left (x \right ) h \left (x \right ) \textit {\_Y} \left (x \right )-\frac {\left (g \left (x \right ) h \left (x \right )+h^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{h \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\frac {d}{d x}\operatorname {DESol}\left (\left \{f \left (x \right ) h \left (x \right ) \textit {\_Y} \left (x \right )-\frac {\left (g \left (x \right ) h \left (x \right )+h^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{h \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{h \left (x \right ) \operatorname {DESol}\left (\left \{f \left (x \right ) h \left (x \right ) \textit {\_Y} \left (x \right )-\frac {\left (g \left (x \right ) h \left (x \right )+h^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{h \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\frac {d}{d x}\operatorname {DESol}\left (\left \{f \left (x \right ) h \left (x \right ) \textit {\_Y} \left (x \right )-\frac {\left (g \left (x \right ) h \left (x \right )+h^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{h \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{h \left (x \right ) \operatorname {DESol}\left (\left \{f \left (x \right ) h \left (x \right ) \textit {\_Y} \left (x \right )-\frac {\left (g \left (x \right ) h \left (x \right )+h^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{h \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\frac {d}{d x}\operatorname {DESol}\left (\left \{f \left (x \right ) h \left (x \right ) \textit {\_Y} \left (x \right )-\frac {\left (g \left (x \right ) h \left (x \right )+h^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{h \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{h \left (x \right ) \operatorname {DESol}\left (\left \{f \left (x \right ) h \left (x \right ) \textit {\_Y} \left (x \right )-\frac {\left (g \left (x \right ) h \left (x \right )+h^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{h \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-g \left (x \right ) y-h \left (x \right ) y^{2}=f \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=f \left (x \right )+g \left (x \right ) y+h \left (x \right ) y^{2} \end {array} \]
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) = (g(x)*h(x)+diff(h(x), x))*(diff(y(x), x))/h(x)-h(x)*f(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)-(h(x)*y(x)^2+y(x)+g(x)*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`
✗ Solution by Maple
dsolve(diff(y(x),x) = f(x)+g(x)*y(x)+h(x)*y(x)^2,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==f[x]+g[x] y[x]+h[x] y[x]^2,y[x],x,IncludeSingularSolutions -> True]
Not solved