Internal problem ID [4401]
Internal file name [OUTPUT/3894_Sunday_June_05_2022_11_36_47_AM_40904880/index.tex]
Book: Differential Equations, By George Boole F.R.S. 1865
Section: Chapter 6
Problem number: 2.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_rational, _Riccati]
\[ \boxed {x y^{\prime }-a y+y^{2}=x^{-\frac {2 a}{3}}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {-a y +y^{2}-x^{-\frac {2 a}{3}}}{x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = \frac {a y}{x}-\frac {y^{2}}{x}+\frac {x^{-\frac {2 a}{3}}}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {x^{-\frac {2 a}{3}}}{x}\), \(f_1(x)=\frac {a}{x}\) and \(f_2(x)=-\frac {1}{x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-\frac {u}{x}} \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 {1}{x^{2}}\\ f_1 f_2 &=-\frac {a}{x^{2}}\\ f_2^2 f_0 &=\frac {x^{-\frac {2 a}{3}}}{x^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -\frac {u^{\prime \prime }\left (x \right )}{x}-\left (\frac {1}{x^{2}}-\frac {a}{x^{2}}\right ) u^{\prime }\left (x \right )+\frac {x^{-\frac {2 a}{3}} u \left (x \right )}{x^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = x^{a} \left (c_{2} \left (3 \sqrt {x^{-\frac {2 a}{3}}}+a \right ) {\mathrm e}^{-\frac {3 x^{-\frac {a}{3}}}{a}}+{\mathrm e}^{\frac {3 x^{-\frac {a}{3}}}{a}} c_{1} \left (-3 \sqrt {x^{-\frac {2 a}{3}}}+a \right )\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {x^{a} \left (c_{2} \left (\left (2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a +x^{-\frac {a}{3}}\right )\right ) {\mathrm e}^{-\frac {3 x^{-\frac {a}{3}}}{a}}+\left (\left (-2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a -x^{-\frac {a}{3}}\right )\right ) {\mathrm e}^{\frac {3 x^{-\frac {a}{3}}}{a}} c_{1} \right )}{x} \] Using the above in (1) gives the solution \[ y = \frac {c_{2} \left (\left (2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a +x^{-\frac {a}{3}}\right )\right ) {\mathrm e}^{-\frac {3 x^{-\frac {a}{3}}}{a}}+\left (\left (-2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a -x^{-\frac {a}{3}}\right )\right ) {\mathrm e}^{\frac {3 x^{-\frac {a}{3}}}{a}} c_{1}}{c_{2} \left (3 \sqrt {x^{-\frac {2 a}{3}}}+a \right ) {\mathrm e}^{-\frac {3 x^{-\frac {a}{3}}}{a}}+{\mathrm e}^{\frac {3 x^{-\frac {a}{3}}}{a}} c_{1} \left (-3 \sqrt {x^{-\frac {2 a}{3}}}+a \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\left (\left (-2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a -x^{-\frac {a}{3}}\right )\right ) c_{3} {\mathrm e}^{\frac {6 x^{-\frac {a}{3}}}{a}}+\left (2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a +x^{-\frac {a}{3}}\right )}{c_{3} \left (-3 \sqrt {x^{-\frac {2 a}{3}}}+a \right ) {\mathrm e}^{\frac {6 x^{-\frac {a}{3}}}{a}}+3 \sqrt {x^{-\frac {2 a}{3}}}+a} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (\left (-2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a -x^{-\frac {a}{3}}\right )\right ) c_{3} {\mathrm e}^{\frac {6 x^{-\frac {a}{3}}}{a}}+\left (2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a +x^{-\frac {a}{3}}\right )}{c_{3} \left (-3 \sqrt {x^{-\frac {2 a}{3}}}+a \right ) {\mathrm e}^{\frac {6 x^{-\frac {a}{3}}}{a}}+3 \sqrt {x^{-\frac {2 a}{3}}}+a} \\ \end{align*}
Verification of solutions
\[ y = \frac {\left (\left (-2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a -x^{-\frac {a}{3}}\right )\right ) c_{3} {\mathrm e}^{\frac {6 x^{-\frac {a}{3}}}{a}}+\left (2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a +x^{-\frac {a}{3}}\right )}{c_{3} \left (-3 \sqrt {x^{-\frac {2 a}{3}}}+a \right ) {\mathrm e}^{\frac {6 x^{-\frac {a}{3}}}{a}}+3 \sqrt {x^{-\frac {2 a}{3}}}+a} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }-a y+y^{2}=x^{-\frac {2 a}{3}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a y-y^{2}+x^{-\frac {2 a}{3}}}{x} \end {array} \]
Maple trace Kovacic algorithm successful
`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-1)*(diff(y(x), x))/x+x^(-1-(2/3)*a)*y(x)/x, y(x)` *** Sublevel 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 -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Group is reducible or imprimitive <- Kovacics algorithm successful <- Equivalence, under non-integer power transformations successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 119
dsolve(x*diff(y(x),x)-a*y(x)+y(x)^2=x^(-2*a/3),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\left (-2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a -x^{-\frac {a}{3}}\right )\right ) {\mathrm e}^{\frac {6 x^{-\frac {a}{3}}}{a}}+\left (\left (2 a +3 x^{-\frac {a}{3}}\right ) \sqrt {x^{-\frac {2 a}{3}}}+a \left (a +x^{-\frac {a}{3}}\right )\right ) c_{1}}{\left (-3 \sqrt {x^{-\frac {2 a}{3}}}+a \right ) {\mathrm e}^{\frac {6 x^{-\frac {a}{3}}}{a}}+c_{1} \left (3 \sqrt {x^{-\frac {2 a}{3}}}+a \right )} \]
✓ Solution by Mathematica
Time used: 0.427 (sec). Leaf size: 270
DSolve[x*y'[x]-a*y[x]+y[x]^2==x^(-2*a/3),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{-a/3} \left (\left (a^2 x^{2 a/3}-3 i a c_1 x^{a/3}+3\right ) \cosh \left (\frac {3 x^{-a/3}}{a}\right )+i \left (a^2 c_1 x^{2 a/3}+3 i a x^{a/3}+3 c_1\right ) \sinh \left (\frac {3 x^{-a/3}}{a}\right )\right )}{\left (a x^{a/3}-3 i c_1\right ) \cosh \left (\frac {3 x^{-a/3}}{a}\right )+i \left (a c_1 x^{a/3}+3 i\right ) \sinh \left (\frac {3 x^{-a/3}}{a}\right )} \\ y(x)\to \frac {\left (a^2 x^{2 a/3}+3\right ) \sinh \left (\frac {3 x^{-a/3}}{a}\right )-3 a x^{a/3} \cosh \left (\frac {3 x^{-a/3}}{a}\right )}{a x^{2 a/3} \sinh \left (\frac {3 x^{-a/3}}{a}\right )-3 x^{a/3} \cosh \left (\frac {3 x^{-a/3}}{a}\right )} \\ \end{align*}