Internal problem ID [8442]
Internal file name [OUTPUT/7375_Sunday_June_05_2022_10_53_52_PM_20378280/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 105.
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]
\[ \boxed {y^{\prime } x +a y^{2} x +b y=-c x -d} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {a x \,y^{2}+b y +c x +d}{x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -a \,y^{2}-\frac {b y}{x}-c -\frac {d}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {c x +d}{x}\), \(f_1(x)=-\frac {b}{x}\) and \(f_2(x)=-a\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-a 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 {b a}{x}\\ f_2^2 f_0 &=-\frac {a^{2} \left (c x +d \right )}{x} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -a u^{\prime \prime }\left (x \right )-\frac {b a u^{\prime }\left (x \right )}{x}-\frac {a^{2} \left (c x +d \right ) u \left (x \right )}{x} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = {\mathrm e}^{-i \sqrt {a}\, \sqrt {c}\, x} \left (\operatorname {KummerU}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{2} +\operatorname {KummerM}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{1} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {\left (-\frac {c_{2} \left (\left (-\frac {1}{2} b^{2}+b \right ) c^{{3}/{2}}+d \left (i \sqrt {a}\, c -\frac {a d \sqrt {c}}{2}\right )\right ) \operatorname {KummerU}\left (\frac {\left (b +2\right ) \sqrt {c}+i \sqrt {a}\, d}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right )}{2}-\frac {c_{1} \left (i c d \sqrt {a}+c^{{3}/{2}} b \right ) \operatorname {KummerM}\left (\frac {\left (b +2\right ) \sqrt {c}+i \sqrt {a}\, d}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right )}{2}+\left (\frac {c^{{3}/{2}} b}{2}+i \left (c x +\frac {d}{2}\right ) \sqrt {a}\, c \right ) \left (\operatorname {KummerU}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{2} +\operatorname {KummerM}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{1} \right )\right ) {\mathrm e}^{-i \sqrt {a}\, \sqrt {c}\, x}}{c^{{3}/{2}} x} \] Using the above in (1) gives the solution \[ y = -\frac {-\frac {c_{2} \left (\left (-\frac {1}{2} b^{2}+b \right ) c^{{3}/{2}}+d \left (i \sqrt {a}\, c -\frac {a d \sqrt {c}}{2}\right )\right ) \operatorname {KummerU}\left (\frac {\left (b +2\right ) \sqrt {c}+i \sqrt {a}\, d}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right )}{2}-\frac {c_{1} \left (i c d \sqrt {a}+c^{{3}/{2}} b \right ) \operatorname {KummerM}\left (\frac {\left (b +2\right ) \sqrt {c}+i \sqrt {a}\, d}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right )}{2}+\left (\frac {c^{{3}/{2}} b}{2}+i \left (c x +\frac {d}{2}\right ) \sqrt {a}\, c \right ) \left (\operatorname {KummerU}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{2} +\operatorname {KummerM}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{1} \right )}{c^{{3}/{2}} x a \left (\operatorname {KummerU}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{2} +\operatorname {KummerM}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{1} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {-\frac {c_{2} \left (\left (-\frac {1}{2} b^{2}+b \right ) c^{{3}/{2}}+d \left (i \sqrt {a}\, c -\frac {a d \sqrt {c}}{2}\right )\right ) \operatorname {KummerU}\left (\frac {\left (b +2\right ) \sqrt {c}+i \sqrt {a}\, d}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right )}{2}-\frac {c_{1} \left (i c d \sqrt {a}+c^{{3}/{2}} b \right ) \operatorname {KummerM}\left (\frac {\left (b +2\right ) \sqrt {c}+i \sqrt {a}\, d}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right )}{2}+\left (\frac {c^{{3}/{2}} b}{2}+i \left (c x +\frac {d}{2}\right ) \sqrt {a}\, c \right ) \left (\operatorname {KummerU}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{2} +\operatorname {KummerM}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{1} \right )}{c^{{3}/{2}} x a \left (\operatorname {KummerU}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{2} +\operatorname {KummerM}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{1} \right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {-\frac {c_{2} \left (\left (-\frac {1}{2} b^{2}+b \right ) c^{{3}/{2}}+d \left (i \sqrt {a}\, c -\frac {a d \sqrt {c}}{2}\right )\right ) \operatorname {KummerU}\left (\frac {\left (b +2\right ) \sqrt {c}+i \sqrt {a}\, d}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right )}{2}-\frac {c_{1} \left (i c d \sqrt {a}+c^{{3}/{2}} b \right ) \operatorname {KummerM}\left (\frac {\left (b +2\right ) \sqrt {c}+i \sqrt {a}\, d}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right )}{2}+\left (\frac {c^{{3}/{2}} b}{2}+i \left (c x +\frac {d}{2}\right ) \sqrt {a}\, c \right ) \left (\operatorname {KummerU}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{2} +\operatorname {KummerM}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{1} \right )}{c^{{3}/{2}} x a \left (\operatorname {KummerU}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{2} +\operatorname {KummerM}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{1} \right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {-\frac {c_{2} \left (\left (-\frac {1}{2} b^{2}+b \right ) c^{{3}/{2}}+d \left (i \sqrt {a}\, c -\frac {a d \sqrt {c}}{2}\right )\right ) \operatorname {KummerU}\left (\frac {\left (b +2\right ) \sqrt {c}+i \sqrt {a}\, d}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right )}{2}-\frac {c_{1} \left (i c d \sqrt {a}+c^{{3}/{2}} b \right ) \operatorname {KummerM}\left (\frac {\left (b +2\right ) \sqrt {c}+i \sqrt {a}\, d}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right )}{2}+\left (\frac {c^{{3}/{2}} b}{2}+i \left (c x +\frac {d}{2}\right ) \sqrt {a}\, c \right ) \left (\operatorname {KummerU}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{2} +\operatorname {KummerM}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{1} \right )}{c^{{3}/{2}} x a \left (\operatorname {KummerU}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{2} +\operatorname {KummerM}\left (\frac {i \sqrt {a}\, d +b \sqrt {c}}{2 \sqrt {c}}, b , 2 i \sqrt {a}\, \sqrt {c}\, x \right ) c_{1} \right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } x +a y^{2} x +b y=-c x -d \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {a y^{2} x +b y+c x +d}{x} \end {array} \]
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*x*y(x)^2 + b*y(x) + c*x + d=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {4 c \left (a \,c^{3} \left (a d -b \sqrt {-a c}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-a c}\, d +c \left (b +2\right )}{2 c}, b +1, 2 x \sqrt {-a c}\right )-\frac {c_{1} \left (a \,d^{2}+b^{2} c \right ) \operatorname {KummerU}\left (\frac {\sqrt {-a c}\, d +c \left (b +2\right )}{2 c}, b +1, 2 x \sqrt {-a c}\right )}{4}+a \,c^{3} \left (b \sqrt {-a c}+a d \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a c}\, d +b c}{2 c}, b +1, 2 x \sqrt {-a c}\right )-\frac {\operatorname {KummerU}\left (\frac {\sqrt {-a c}\, d +b c}{2 c}, b +1, 2 x \sqrt {-a c}\right ) c_{1} \left (b c -\sqrt {-a c}\, d \right )}{2}\right )}{4 a^{2} c^{3} \left (\sqrt {-a c}\, d +b c \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a c}\, d +c \left (b +2\right )}{2 c}, b +1, 2 x \sqrt {-a c}\right )-c_{1} \sqrt {-a c}\, \left (a \,d^{2}+b^{2} c \right ) \operatorname {KummerU}\left (\frac {\sqrt {-a c}\, d +c \left (b +2\right )}{2 c}, b +1, 2 x \sqrt {-a c}\right )+4 c \left (a^{2} c^{2} \left (b c -\sqrt {-a c}\, d \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a c}\, d +b c}{2 c}, b +1, 2 x \sqrt {-a c}\right )+\frac {\operatorname {KummerU}\left (\frac {\sqrt {-a c}\, d +b c}{2 c}, b +1, 2 x \sqrt {-a c}\right ) c_{1} \left (b \sqrt {-a c}+a d \right )}{2}\right )} \]
✓ Solution by Mathematica
Time used: 0.453 (sec). Leaf size: 421
DSolve[x*y'[x] + a*x*y[x]^2 + b*y[x] + c*x + d==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i \left (\sqrt {c} c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (b+\frac {i \sqrt {a} d}{\sqrt {c}}\right ),b,2 i \sqrt {a} \sqrt {c} x\right )+c_1 \left (b \sqrt {c}+i \sqrt {a} d\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (b+\frac {i \sqrt {a} d}{\sqrt {c}}+2\right ),b+1,2 i \sqrt {a} \sqrt {c} x\right )+\sqrt {c} \left (2 L_{-\frac {b}{2}-\frac {i \sqrt {a} d}{2 \sqrt {c}}-1}^b\left (2 i \sqrt {a} \sqrt {c} x\right )+L_{-\frac {b}{2}-\frac {i \sqrt {a} d}{2 \sqrt {c}}}^{b-1}\left (2 i \sqrt {a} \sqrt {c} x\right )\right )\right )}{\sqrt {a} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (b+\frac {i \sqrt {a} d}{\sqrt {c}}\right ),b,2 i \sqrt {a} \sqrt {c} x\right )+L_{-\frac {b}{2}-\frac {i \sqrt {a} d}{2 \sqrt {c}}}^{b-1}\left (2 i \sqrt {a} \sqrt {c} x\right )\right )} \\ y(x)\to \frac {\frac {\left (\sqrt {a} d-i b \sqrt {c}\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (b+\frac {i \sqrt {a} d}{\sqrt {c}}+2\right ),b+1,2 i \sqrt {a} \sqrt {c} x\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (b+\frac {i \sqrt {a} d}{\sqrt {c}}\right ),b,2 i \sqrt {a} \sqrt {c} x\right )}-i \sqrt {c}}{\sqrt {a}} \\ \end{align*}