Internal problem ID [8443]
Internal file name [OUTPUT/7376_Sunday_June_05_2022_10_53_54_PM_86521446/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 106.
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]
\[ \boxed {x y^{\prime }+x^{a} y^{2}+\frac {\left (a -b \right ) y}{2}=-x^{b}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {2 x^{a} y^{2}+a y -b y +2 x^{b}}{2 x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {x^{a} y^{2}}{x}-\frac {a y}{2 x}+\frac {b y}{2 x}-\frac {x^{b}}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {x^{b}}{x}\), \(f_1(x)=-\frac {a -b}{2 x}\) and \(f_2(x)=-\frac {x^{a}}{x}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-\frac {x^{a} 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 {x^{a}}{x^{2}}-\frac {x^{a} a}{x^{2}}\\ f_1 f_2 &=\frac {\left (a -b \right ) x^{a}}{2 x^{2}}\\ f_2^2 f_0 &=-\frac {x^{2 a} x^{b}}{x^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -\frac {x^{a} u^{\prime \prime }\left (x \right )}{x}-\left (\frac {x^{a}}{x^{2}}-\frac {x^{a} a}{x^{2}}+\frac {\left (a -b \right ) x^{a}}{2 x^{2}}\right ) u^{\prime }\left (x \right )-\frac {x^{2 a} x^{b} u \left (x \right )}{x^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{1} \sin \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )+c_{2} \cos \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\sqrt {x^{a +b}}\, \left (c_{1} \cos \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )-c_{2} \sin \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )\right )}{x} \] Using the above in (1) gives the solution \[ y = \frac {\sqrt {x^{a +b}}\, \left (c_{1} \cos \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )-c_{2} \sin \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )\right ) x^{-a}}{c_{1} \sin \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )+c_{2} \cos \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\sqrt {x^{a +b}}\, \left (c_{3} \cos \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )-\sin \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )\right ) x^{-a}}{c_{3} \sin \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )+\cos \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sqrt {x^{a +b}}\, \left (c_{3} \cos \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )-\sin \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )\right ) x^{-a}}{c_{3} \sin \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )+\cos \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {x^{a +b}}\, \left (c_{3} \cos \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )-\sin \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )\right ) x^{-a}}{c_{3} \sin \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )+\cos \left (\frac {2 \sqrt {x^{a +b}}}{a +b}\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }+x^{a} y^{2}+\frac {\left (a -b \right ) y}{2}=-x^{b} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-x^{a} y^{2}-\frac {\left (a -b \right ) y}{2}-x^{b}}{x} \end {array} \]
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 <- Chini successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 39
dsolve(x*diff(y(x),x) + x^a*y(x)^2 + (a-b)*y(x)/2 + x^b=0,y(x), singsol=all)
\[ y \left (x \right ) = -\tan \left (\frac {2 x^{\frac {a}{2}+\frac {b}{2}}+c_{1} \left (a +b \right )}{a +b}\right ) x^{-\frac {a}{2}+\frac {b}{2}} \]
✓ Solution by Mathematica
Time used: 0.539 (sec). Leaf size: 40
DSolve[x*y'[x] + x^a*y[x]^2 + (a-b)*y[x]/2 + x^b==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -x^{\frac {b-a}{2}} \tan \left (\frac {2 x^{\frac {a+b}{2}}}{a+b}-c_1\right ) \]