Internal problem ID [8478]
Internal file name [OUTPUT/7411_Sunday_June_05_2022_10_54_40_PM_27372071/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 142.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[_rational, _Riccati]
\[ \boxed {x^{2} \left (y^{\prime }-y^{2}\right )-a \,x^{2} y=-a x -2} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {a \,x^{2} y +y^{2} x^{2}-a x -2}{x^{2}} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = y a +y^{2}-\frac {a}{x}-\frac {2}{x^{2}} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {-a x -2}{x^{2}}\), \(f_1(x)=a\) and \(f_2(x)=1\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{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\\ f_2^2 f_0 &=\frac {-a x -2}{x^{2}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} u^{\prime \prime }\left (x \right )-a u^{\prime }\left (x \right )+\frac {\left (-a x -2\right ) u \left (x \right )}{x^{2}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = \frac {\left (a^{2} x^{2}-2 a x +2\right ) {\mathrm e}^{a x} c_{1} +c_{2}}{x} \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {c_{1} \left (a x -1\right ) \left (a^{2} x^{2}+2\right ) {\mathrm e}^{a x}-c_{2}}{x^{2}} \] Using the above in (1) gives the solution \[ y = -\frac {c_{1} \left (a x -1\right ) \left (a^{2} x^{2}+2\right ) {\mathrm e}^{a x}-c_{2}}{x \left (\left (a^{2} x^{2}-2 a x +2\right ) {\mathrm e}^{a x} c_{1} +c_{2} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {-c_{3} \left (a x -1\right ) \left (a^{2} x^{2}+2\right ) {\mathrm e}^{a x}+1}{x \left (\left (a^{2} x^{2}-2 a x +2\right ) {\mathrm e}^{a x} c_{3} +1\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-c_{3} \left (a x -1\right ) \left (a^{2} x^{2}+2\right ) {\mathrm e}^{a x}+1}{x \left (\left (a^{2} x^{2}-2 a x +2\right ) {\mathrm e}^{a x} c_{3} +1\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {-c_{3} \left (a x -1\right ) \left (a^{2} x^{2}+2\right ) {\mathrm e}^{a x}+1}{x \left (\left (a^{2} x^{2}-2 a x +2\right ) {\mathrm e}^{a x} c_{3} +1\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (y^{\prime }-y^{2}\right )-a \,x^{2} y=-a x -2 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{2} x^{2}+a \,x^{2} y-a x -2}{x^{2}} \end {array} \]
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 to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = (diff(y(x), x))*a+(a*x+2)*y(x)/x^2, y(x)` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] <- linear_1 successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 52
dsolve(x^2*(diff(y(x),x)-y(x)^2) - a*x^2*y(x) + a*x + 2=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-\left (a x -1\right ) \left (a^{2} x^{2}+2\right ) {\mathrm e}^{a x}+c_{1}}{\left (\left (a^{2} x^{2}-2 a x +2\right ) {\mathrm e}^{a x}+c_{1} \right ) x} \]
✓ Solution by Mathematica
Time used: 0.375 (sec). Leaf size: 78
DSolve[x^2*(y'[x]-y[x]^2) - a*x^2*y[x] + a*x + 2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{a x} \left (-a^3 x^3+a^2 x^2-2 a x+2\right )+a^3 c_1}{x \left (e^{a x} \left (a^2 x^2-2 a x+2\right )+a^3 c_1\right )} \\ y(x)\to \frac {1}{x} \\ \end{align*}