Internal problem ID [10367]
Internal file name [OUTPUT/9315_Monday_June_06_2022_01_51_16_PM_89246891/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power
Functions
Problem number: 38.
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]
\[ \boxed {y^{\prime } x +a_{3} x y^{2}+a_{2} y=-a_{1} x -a_{0}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {a_{3} x \,y^{2}+a_{1} x +a_{2} y +a_{0}}{x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -a_{3} y^{2}-a_{1} -\frac {a_{2} y}{x}-\frac {a_{0}}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {a_{1} x +a_{0}}{x}\), \(f_1(x)=-\frac {a_{2}}{x}\) and \(f_2(x)=-a_{3}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-a_{3} 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 &=\frac {a_{2} a_{3}}{x}\\ f_2^2 f_0 &=-\frac {a_{3}^{2} \left (a_{1} x +a_{0} \right )}{x} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -a_{3} u^{\prime \prime }\left (x \right )-\frac {a_{2} a_{3} u^{\prime }\left (x \right )}{x}-\frac {a_{3}^{2} \left (a_{1} x +a_{0} \right ) u \left (x \right )}{x} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = {\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \left (\operatorname {KummerM}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{1} +\operatorname {KummerU}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{2} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {{\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \left (-\frac {\left (\left (-\frac {1}{2} a_{2}^{2}+a_{2} \right ) a_{1}^{\frac {3}{2}}+\left (i a_{1} \sqrt {a_{3}}-\frac {a_{0} a_{3} \sqrt {a_{1}}}{2}\right ) a_{0} \right ) c_{2} \operatorname {KummerU}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i a_{0} \sqrt {a_{3}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )}{2}-\frac {c_{1} \left (i a_{1} \sqrt {a_{3}}\, a_{0} +a_{2} a_{1}^{\frac {3}{2}}\right ) \operatorname {KummerM}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i a_{0} \sqrt {a_{3}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )}{2}+\left (\frac {a_{2} a_{1}^{\frac {3}{2}}}{2}+i \sqrt {a_{3}}\, \left (a_{1} x +\frac {a_{0}}{2}\right ) a_{1} \right ) \left (\operatorname {KummerM}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{1} +\operatorname {KummerU}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{2} \right )\right )}{a_{1}^{\frac {3}{2}} x} \] Using the above in (1) gives the solution \[ y = -\frac {-\frac {\left (\left (-\frac {1}{2} a_{2}^{2}+a_{2} \right ) a_{1}^{\frac {3}{2}}+\left (i a_{1} \sqrt {a_{3}}-\frac {a_{0} a_{3} \sqrt {a_{1}}}{2}\right ) a_{0} \right ) c_{2} \operatorname {KummerU}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i a_{0} \sqrt {a_{3}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )}{2}-\frac {c_{1} \left (i a_{1} \sqrt {a_{3}}\, a_{0} +a_{2} a_{1}^{\frac {3}{2}}\right ) \operatorname {KummerM}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i a_{0} \sqrt {a_{3}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )}{2}+\left (\frac {a_{2} a_{1}^{\frac {3}{2}}}{2}+i \sqrt {a_{3}}\, \left (a_{1} x +\frac {a_{0}}{2}\right ) a_{1} \right ) \left (\operatorname {KummerM}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{1} +\operatorname {KummerU}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{2} \right )}{a_{1}^{\frac {3}{2}} x a_{3} \left (\operatorname {KummerM}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{1} +\operatorname {KummerU}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{2} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = \frac {\frac {\left (-\left (\frac {1}{2} a_{2}^{2}-a_{2} \right ) a_{1}^{\frac {3}{2}}+i a_{1} \sqrt {a_{3}}\, a_{0} -\frac {a_{0}^{2} a_{3} \sqrt {a_{1}}}{2}\right ) \operatorname {KummerU}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i a_{0} \sqrt {a_{3}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )}{2}+\frac {c_{3} \left (i a_{1} \sqrt {a_{3}}\, a_{0} +a_{2} a_{1}^{\frac {3}{2}}\right ) \operatorname {KummerM}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i a_{0} \sqrt {a_{3}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )}{2}-\left (\frac {a_{2} a_{1}^{\frac {3}{2}}}{2}+i \sqrt {a_{3}}\, \left (a_{1} x +\frac {a_{0}}{2}\right ) a_{1} \right ) \left (\operatorname {KummerM}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )}{a_{1}^{\frac {3}{2}} x a_{3} \left (\operatorname {KummerM}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\frac {\left (-\left (\frac {1}{2} a_{2}^{2}-a_{2} \right ) a_{1}^{\frac {3}{2}}+i a_{1} \sqrt {a_{3}}\, a_{0} -\frac {a_{0}^{2} a_{3} \sqrt {a_{1}}}{2}\right ) \operatorname {KummerU}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i a_{0} \sqrt {a_{3}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )}{2}+\frac {c_{3} \left (i a_{1} \sqrt {a_{3}}\, a_{0} +a_{2} a_{1}^{\frac {3}{2}}\right ) \operatorname {KummerM}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i a_{0} \sqrt {a_{3}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )}{2}-\left (\frac {a_{2} a_{1}^{\frac {3}{2}}}{2}+i \sqrt {a_{3}}\, \left (a_{1} x +\frac {a_{0}}{2}\right ) a_{1} \right ) \left (\operatorname {KummerM}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )}{a_{1}^{\frac {3}{2}} x a_{3} \left (\operatorname {KummerM}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\frac {\left (-\left (\frac {1}{2} a_{2}^{2}-a_{2} \right ) a_{1}^{\frac {3}{2}}+i a_{1} \sqrt {a_{3}}\, a_{0} -\frac {a_{0}^{2} a_{3} \sqrt {a_{1}}}{2}\right ) \operatorname {KummerU}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i a_{0} \sqrt {a_{3}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )}{2}+\frac {c_{3} \left (i a_{1} \sqrt {a_{3}}\, a_{0} +a_{2} a_{1}^{\frac {3}{2}}\right ) \operatorname {KummerM}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i a_{0} \sqrt {a_{3}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )}{2}-\left (\frac {a_{2} a_{1}^{\frac {3}{2}}}{2}+i \sqrt {a_{3}}\, \left (a_{1} x +\frac {a_{0}}{2}\right ) a_{1} \right ) \left (\operatorname {KummerM}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )}{a_{1}^{\frac {3}{2}} x a_{3} \left (\operatorname {KummerM}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i a_{0} \sqrt {a_{3}}+a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } x +a_{3} x y^{2}+a_{2} y=-a_{1} x -a_{0} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {a_{3} x y^{2}+a_{2} y+a_{1} x +a_{0}}{x} \end {array} \]
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 differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying Riccati trying Riccati sub-methods: <- Abel AIR successful: ODE belongs to the 1F1 2-parameter class`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 403
dsolve(x*diff(y(x),x)+a__3*x*y(x)^2+a__2*y(x)+a__1*x+a__0=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {4 a_{1} \left (a_{1}^{3} a_{3} \left (a_{3} a_{0} -a_{2} \sqrt {-a_{1} a_{3}}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )-\frac {c_{1} \left (a_{0}^{2} a_{3} +a_{1} a_{2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )}{4}+a_{1}^{3} a_{3} \left (a_{2} \sqrt {-a_{1} a_{3}}+a_{3} a_{0} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )+\frac {\operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right ) c_{1} \left (\sqrt {-a_{1} a_{3}}\, a_{0} -a_{1} a_{2} \right )}{2}\right )}{4 a_{1}^{3} a_{3}^{2} \left (\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )-c_{1} \sqrt {-a_{1} a_{3}}\, \left (a_{0}^{2} a_{3} +a_{1} a_{2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )+2 a_{1} \left (-2 a_{1}^{2} a_{3}^{2} \left (\sqrt {-a_{1} a_{3}}\, a_{0} -a_{1} a_{2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )+\operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right ) c_{1} \left (a_{2} \sqrt {-a_{1} a_{3}}+a_{3} a_{0} \right )\right )} \]
✓ Solution by Mathematica
Time used: 0.748 (sec). Leaf size: 541
DSolve[x*y'[x]+a3*x*y[x]^2+a2*y[x]+a1*x+a0==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i \left (\sqrt {\text {a1}} c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+c_1 \left (\sqrt {\text {a1}} \text {a2}+i \text {a0} \sqrt {\text {a3}}\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}+2\right ),\text {a2}+1,2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+\sqrt {\text {a1}} \left (2 L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}-1}^{\text {a2}}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}}^{\text {a2}-1}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )\right )\right )}{\sqrt {\text {a3}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}}^{\text {a2}-1}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )\right )} \\ y(x)\to \frac {\frac {\left (\text {a0} \sqrt {\text {a3}}-i \sqrt {\text {a1}} \text {a2}\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}+2\right ),\text {a2}+1,2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}-i \sqrt {\text {a1}}}{\sqrt {\text {a3}}} \\ y(x)\to \frac {\frac {\left (\text {a0} \sqrt {\text {a3}}-i \sqrt {\text {a1}} \text {a2}\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}+2\right ),\text {a2}+1,2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}-i \sqrt {\text {a1}}}{\sqrt {\text {a3}}} \\ \end{align*}