Internal problem ID [8370]
Internal file name [OUTPUT/7303_Sunday_June_05_2022_05_43_51_PM_5137925/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 33.
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]
\[ \boxed {y^{\prime }-\frac {y^{2} f^{\prime }\left (x \right )}{g \left (x \right )}=-\frac {g^{\prime }\left (x \right )}{f \left (x \right )}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {y^{2} f^{\prime }\left (x \right ) f \left (x \right )-g^{\prime }\left (x \right ) g \left (x \right )}{g \left (x \right ) f \left (x \right )} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {y^{2} f^{\prime }\left (x \right )}{g \left (x \right )}-\frac {g^{\prime }\left (x \right )}{f \left (x \right )} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {g^{\prime }\left (x \right )}{f \left (x \right )}\), \(f_1(x)=0\) and \(f_2(x)=\frac {f^{\prime }\left (x \right )}{g \left (x \right )}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{\frac {f^{\prime }\left (x \right ) u}{g \left (x \right )}} \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 {f^{\prime \prime }\left (x \right )}{g \left (x \right )}-\frac {f^{\prime }\left (x \right ) g^{\prime }\left (x \right )}{g \left (x \right )^{2}}\\ f_1 f_2 &=0\\ f_2^2 f_0 &=-\frac {{f^{\prime }\left (x \right )}^{2} g^{\prime }\left (x \right )}{g \left (x \right )^{2} f \left (x \right )} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} \frac {f^{\prime }\left (x \right ) u^{\prime \prime }\left (x \right )}{g \left (x \right )}-\left (\frac {f^{\prime \prime }\left (x \right )}{g \left (x \right )}-\frac {f^{\prime }\left (x \right ) g^{\prime }\left (x \right )}{g \left (x \right )^{2}}\right ) u^{\prime }\left (x \right )-\frac {{f^{\prime }\left (x \right )}^{2} g^{\prime }\left (x \right ) u \left (x \right )}{g \left (x \right )^{2} f \left (x \right )} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {\left (\frac {f^{\prime \prime }\left (x \right )}{g \left (x \right )}-\frac {f^{\prime }\left (x \right ) g^{\prime }\left (x \right )}{g \left (x \right )^{2}}\right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )}{f^{\prime }\left (x \right )}-\frac {f^{\prime }\left (x \right ) g^{\prime }\left (x \right ) \textit {\_Y} \left (x \right )}{g \left (x \right ) f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {d}{d x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {\left (\frac {f^{\prime \prime }\left (x \right )}{g \left (x \right )}-\frac {f^{\prime }\left (x \right ) g^{\prime }\left (x \right )}{g \left (x \right )^{2}}\right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )}{f^{\prime }\left (x \right )}-\frac {f^{\prime }\left (x \right ) g^{\prime }\left (x \right ) \textit {\_Y} \left (x \right )}{g \left (x \right ) f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right ) \] Using the above in (1) gives the solution \[ y = -\frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {\left (\frac {f^{\prime \prime }\left (x \right )}{g \left (x \right )}-\frac {f^{\prime }\left (x \right ) g^{\prime }\left (x \right )}{g \left (x \right )^{2}}\right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )}{f^{\prime }\left (x \right )}-\frac {f^{\prime }\left (x \right ) g^{\prime }\left (x \right ) \textit {\_Y} \left (x \right )}{g \left (x \right ) f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) g \left (x \right )}{f^{\prime }\left (x \right ) \operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )-\frac {\left (\frac {f^{\prime \prime }\left (x \right )}{g \left (x \right )}-\frac {f^{\prime }\left (x \right ) g^{\prime }\left (x \right )}{g \left (x \right )^{2}}\right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )}{f^{\prime }\left (x \right )}-\frac {f^{\prime }\left (x \right ) g^{\prime }\left (x \right ) \textit {\_Y} \left (x \right )}{g \left (x \right ) f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) g \left (x \right ) f \left (x \right ) f^{\prime }\left (x \right )-f^{\prime \prime }\left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right ) f \left (x \right )-{f^{\prime }\left (x \right )}^{2} g^{\prime }\left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) g^{\prime }\left (x \right ) \textit {\_Y}^{\prime }\left (x \right ) f \left (x \right )}{g \left (x \right ) f^{\prime }\left (x \right ) f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) g \left (x \right )}{f^{\prime }\left (x \right ) \operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) g \left (x \right ) f \left (x \right ) f^{\prime }\left (x \right )-f^{\prime \prime }\left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right ) f \left (x \right )-{f^{\prime }\left (x \right )}^{2} g^{\prime }\left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) g^{\prime }\left (x \right ) \textit {\_Y}^{\prime }\left (x \right ) f \left (x \right )}{g \left (x \right ) f^{\prime }\left (x \right ) f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) g \left (x \right ) f \left (x \right ) f^{\prime }\left (x \right )-f^{\prime \prime }\left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right ) f \left (x \right )-{f^{\prime }\left (x \right )}^{2} g^{\prime }\left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) g^{\prime }\left (x \right ) \textit {\_Y}^{\prime }\left (x \right ) f \left (x \right )}{g \left (x \right ) f^{\prime }\left (x \right ) f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) g \left (x \right )}{f^{\prime }\left (x \right ) \operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) g \left (x \right ) f \left (x \right ) f^{\prime }\left (x \right )-f^{\prime \prime }\left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right ) f \left (x \right )-{f^{\prime }\left (x \right )}^{2} g^{\prime }\left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) g^{\prime }\left (x \right ) \textit {\_Y}^{\prime }\left (x \right ) f \left (x \right )}{g \left (x \right ) f^{\prime }\left (x \right ) f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) g \left (x \right ) f \left (x \right ) f^{\prime }\left (x \right )-f^{\prime \prime }\left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right ) f \left (x \right )-{f^{\prime }\left (x \right )}^{2} g^{\prime }\left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) g^{\prime }\left (x \right ) \textit {\_Y}^{\prime }\left (x \right ) f \left (x \right )}{g \left (x \right ) f^{\prime }\left (x \right ) f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) g \left (x \right )}{f^{\prime }\left (x \right ) \operatorname {DESol}\left (\left \{\frac {\textit {\_Y}^{\prime \prime }\left (x \right ) g \left (x \right ) f \left (x \right ) f^{\prime }\left (x \right )-f^{\prime \prime }\left (x \right ) g \left (x \right ) \textit {\_Y}^{\prime }\left (x \right ) f \left (x \right )-{f^{\prime }\left (x \right )}^{2} g^{\prime }\left (x \right ) \textit {\_Y} \left (x \right )+f^{\prime }\left (x \right ) g^{\prime }\left (x \right ) \textit {\_Y}^{\prime }\left (x \right ) f \left (x \right )}{g \left (x \right ) f^{\prime }\left (x \right ) f \left (x \right )}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{2} f^{\prime }\left (x \right ) f \left (x \right )-y^{\prime } g \left (x \right ) f \left (x \right )-g^{\prime }\left (x \right ) g \left (x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-y^{2} f^{\prime }\left (x \right ) f \left (x \right )+g^{\prime }\left (x \right ) g \left (x \right )}{g \left (x \right ) f \left (x \right )} \end {array} \]
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) = -((diff(f(x), x))*(diff(g(x), x))-(diff(diff(f(x), x), x))*g(x))*(diff Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying 2nd order exact linear trying symmetries linear in x and y(x) trying to convert to a linear ODE with constant coefficients trying 2nd order, integrating factor of the form mu(x,y) trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler <- unable to find a useful change of variables trying a symmetry of the form [xi=0, eta=F(x)] trying to convert to an ODE of Bessel type -> Trying a change of variables to reduce to Bernoulli -> Calling odsolve with the ODE`, diff(y(x), x)-((diff(f(x), x))*y(x)^2/g(x)+y(x)-x^2*(diff(g(x), x))/f(x))/x, y(x), explicit` 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`[0, f(x)^2*(y+g(x)/f(x))^2]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 58
dsolve(diff(y(x),x) - y(x)^2*diff(f(x),x)/g(x) + diff(g(x),x)/f(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-g \left (x \right ) f \left (x \right ) \left (\int \frac {\frac {d}{d x}f \left (x \right )}{g \left (x \right ) f \left (x \right )^{2}}d x \right )-g \left (x \right ) f \left (x \right ) c_{1} -1}{f \left (x \right )^{2} \left (\int \frac {\frac {d}{d x}f \left (x \right )}{g \left (x \right ) f \left (x \right )^{2}}d x +c_{1} \right )} \]
✓ Solution by Mathematica
Time used: 0.353 (sec). Leaf size: 160
DSolve[y'[x] - y[x]^2*f'[x]/g[x] + g'[x]/f[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{(g(x)+f(x) K[2])^2}-\int _1^x\left (\frac {2 \left (f(K[1]) K[2]^2 f'(K[1])-g(K[1]) g'(K[1])\right )}{g(K[1]) (g(K[1])+f(K[1]) K[2])^3}-\frac {2 K[2] f'(K[1])}{g(K[1]) (g(K[1])+f(K[1]) K[2])^2}\right )dK[1]\right )dK[2]+\int _1^x-\frac {f(K[1]) y(x)^2 f'(K[1])-g(K[1]) g'(K[1])}{f(K[1]) g(K[1]) (g(K[1])+f(K[1]) y(x))^2}dK[1]=c_1,y(x)\right ] \]