Internal problem ID [3435]
Internal file name [OUTPUT/2928_Sunday_June_05_2022_08_47_16_AM_52953187/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 7
Problem number: 179.
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]
\[ \boxed {x y^{\prime }+\left (\operatorname {a2} +\operatorname {a3} x y\right ) y=-\operatorname {a1} x -\operatorname {a0}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {\operatorname {a3} x \,y^{2}+\operatorname {a1} x +\operatorname {a2} y +\operatorname {a0}}{x} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\operatorname {a3} \,y^{2}-\operatorname {a1} -\frac {\operatorname {a2} y}{x}-\frac {\operatorname {a0}}{x} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {\operatorname {a1} x +\operatorname {a0}}{x}\), \(f_1(x)=-\frac {\operatorname {a2}}{x}\) and \(f_2(x)=-\operatorname {a3}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-\operatorname {a3} 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 {\operatorname {a2} \operatorname {a3}}{x}\\ f_2^2 f_0 &=-\frac {\operatorname {a3}^{2} \left (\operatorname {a1} x +\operatorname {a0} \right )}{x} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -\operatorname {a3} u^{\prime \prime }\left (x \right )-\frac {\operatorname {a2} \operatorname {a3} u^{\prime }\left (x \right )}{x}-\frac {\operatorname {a3}^{2} \left (\operatorname {a1} x +\operatorname {a0} \right ) u \left (x \right )}{x} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = {\mathrm e}^{-i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x} \left (\operatorname {KummerM}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right ) c_{1} +\operatorname {KummerU}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right ) c_{2} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {{\mathrm e}^{-i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x} \left (-\frac {\left (\left (-\frac {1}{2} \operatorname {a2}^{2}+\operatorname {a2} \right ) \operatorname {a1}^{\frac {3}{2}}+\operatorname {a0} \left (i \sqrt {\operatorname {a3}}\, \operatorname {a1} -\frac {\operatorname {a0} \operatorname {a3} \sqrt {\operatorname {a1}}}{2}\right )\right ) c_{2} \operatorname {KummerU}\left (\frac {\left (\operatorname {a2} +2\right ) \sqrt {\operatorname {a1}}+i \operatorname {a0} \sqrt {\operatorname {a3}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )}{2}-\frac {\left (i \operatorname {a1} \sqrt {\operatorname {a3}}\, \operatorname {a0} +\operatorname {a1}^{\frac {3}{2}} \operatorname {a2} \right ) c_{1} \operatorname {KummerM}\left (\frac {\left (\operatorname {a2} +2\right ) \sqrt {\operatorname {a1}}+i \operatorname {a0} \sqrt {\operatorname {a3}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )}{2}+\left (\frac {\operatorname {a1}^{\frac {3}{2}} \operatorname {a2}}{2}+i \sqrt {\operatorname {a3}}\, \operatorname {a1} \left (\operatorname {a1} x +\frac {\operatorname {a0}}{2}\right )\right ) \left (\operatorname {KummerM}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right ) c_{1} +\operatorname {KummerU}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right ) c_{2} \right )\right )}{\operatorname {a1}^{\frac {3}{2}} x} \] Using the above in (1) gives the solution \[ y = -\frac {-\frac {\left (\left (-\frac {1}{2} \operatorname {a2}^{2}+\operatorname {a2} \right ) \operatorname {a1}^{\frac {3}{2}}+\operatorname {a0} \left (i \sqrt {\operatorname {a3}}\, \operatorname {a1} -\frac {\operatorname {a0} \operatorname {a3} \sqrt {\operatorname {a1}}}{2}\right )\right ) c_{2} \operatorname {KummerU}\left (\frac {\left (\operatorname {a2} +2\right ) \sqrt {\operatorname {a1}}+i \operatorname {a0} \sqrt {\operatorname {a3}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )}{2}-\frac {\left (i \operatorname {a1} \sqrt {\operatorname {a3}}\, \operatorname {a0} +\operatorname {a1}^{\frac {3}{2}} \operatorname {a2} \right ) c_{1} \operatorname {KummerM}\left (\frac {\left (\operatorname {a2} +2\right ) \sqrt {\operatorname {a1}}+i \operatorname {a0} \sqrt {\operatorname {a3}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )}{2}+\left (\frac {\operatorname {a1}^{\frac {3}{2}} \operatorname {a2}}{2}+i \sqrt {\operatorname {a3}}\, \operatorname {a1} \left (\operatorname {a1} x +\frac {\operatorname {a0}}{2}\right )\right ) \left (\operatorname {KummerM}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right ) c_{1} +\operatorname {KummerU}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right ) c_{2} \right )}{\operatorname {a1}^{\frac {3}{2}} x \operatorname {a3} \left (\operatorname {KummerM}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right ) c_{1} +\operatorname {KummerU}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, 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} \operatorname {a2}^{2}-\operatorname {a2} \right ) \operatorname {a1}^{\frac {3}{2}}+i \operatorname {a1} \sqrt {\operatorname {a3}}\, \operatorname {a0} -\frac {\operatorname {a0}^{2} \operatorname {a3} \sqrt {\operatorname {a1}}}{2}\right ) \operatorname {KummerU}\left (\frac {\left (\operatorname {a2} +2\right ) \sqrt {\operatorname {a1}}+i \operatorname {a0} \sqrt {\operatorname {a3}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )}{2}+\frac {\left (i \operatorname {a1} \sqrt {\operatorname {a3}}\, \operatorname {a0} +\operatorname {a1}^{\frac {3}{2}} \operatorname {a2} \right ) c_{3} \operatorname {KummerM}\left (\frac {\left (\operatorname {a2} +2\right ) \sqrt {\operatorname {a1}}+i \operatorname {a0} \sqrt {\operatorname {a3}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )}{2}-\left (\frac {\operatorname {a1}^{\frac {3}{2}} \operatorname {a2}}{2}+i \sqrt {\operatorname {a3}}\, \operatorname {a1} \left (\operatorname {a1} x +\frac {\operatorname {a0}}{2}\right )\right ) \left (\operatorname {KummerM}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )\right )}{\operatorname {a1}^{\frac {3}{2}} x \operatorname {a3} \left (\operatorname {KummerM}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\frac {\left (-\left (\frac {1}{2} \operatorname {a2}^{2}-\operatorname {a2} \right ) \operatorname {a1}^{\frac {3}{2}}+i \operatorname {a1} \sqrt {\operatorname {a3}}\, \operatorname {a0} -\frac {\operatorname {a0}^{2} \operatorname {a3} \sqrt {\operatorname {a1}}}{2}\right ) \operatorname {KummerU}\left (\frac {\left (\operatorname {a2} +2\right ) \sqrt {\operatorname {a1}}+i \operatorname {a0} \sqrt {\operatorname {a3}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )}{2}+\frac {\left (i \operatorname {a1} \sqrt {\operatorname {a3}}\, \operatorname {a0} +\operatorname {a1}^{\frac {3}{2}} \operatorname {a2} \right ) c_{3} \operatorname {KummerM}\left (\frac {\left (\operatorname {a2} +2\right ) \sqrt {\operatorname {a1}}+i \operatorname {a0} \sqrt {\operatorname {a3}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )}{2}-\left (\frac {\operatorname {a1}^{\frac {3}{2}} \operatorname {a2}}{2}+i \sqrt {\operatorname {a3}}\, \operatorname {a1} \left (\operatorname {a1} x +\frac {\operatorname {a0}}{2}\right )\right ) \left (\operatorname {KummerM}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )\right )}{\operatorname {a1}^{\frac {3}{2}} x \operatorname {a3} \left (\operatorname {KummerM}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\frac {\left (-\left (\frac {1}{2} \operatorname {a2}^{2}-\operatorname {a2} \right ) \operatorname {a1}^{\frac {3}{2}}+i \operatorname {a1} \sqrt {\operatorname {a3}}\, \operatorname {a0} -\frac {\operatorname {a0}^{2} \operatorname {a3} \sqrt {\operatorname {a1}}}{2}\right ) \operatorname {KummerU}\left (\frac {\left (\operatorname {a2} +2\right ) \sqrt {\operatorname {a1}}+i \operatorname {a0} \sqrt {\operatorname {a3}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )}{2}+\frac {\left (i \operatorname {a1} \sqrt {\operatorname {a3}}\, \operatorname {a0} +\operatorname {a1}^{\frac {3}{2}} \operatorname {a2} \right ) c_{3} \operatorname {KummerM}\left (\frac {\left (\operatorname {a2} +2\right ) \sqrt {\operatorname {a1}}+i \operatorname {a0} \sqrt {\operatorname {a3}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )}{2}-\left (\frac {\operatorname {a1}^{\frac {3}{2}} \operatorname {a2}}{2}+i \sqrt {\operatorname {a3}}\, \operatorname {a1} \left (\operatorname {a1} x +\frac {\operatorname {a0}}{2}\right )\right ) \left (\operatorname {KummerM}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )\right )}{\operatorname {a1}^{\frac {3}{2}} x \operatorname {a3} \left (\operatorname {KummerM}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right ) c_{3} +\operatorname {KummerU}\left (\frac {i \operatorname {a0} \sqrt {\operatorname {a3}}+\operatorname {a2} \sqrt {\operatorname {a1}}}{2 \sqrt {\operatorname {a1}}}, \operatorname {a2} , 2 i \sqrt {\operatorname {a1}}\, \sqrt {\operatorname {a3}}\, x \right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y^{\prime }+\left (\mathit {a2} +\mathit {a3} x y\right ) y=-\mathit {a1} x -\mathit {a0} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {\mathit {a0} +\mathit {a1} x +\left (\mathit {a2} +\mathit {a3} x y\right ) y}{x} \end {array} \]
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)+a0+a1*x+(a2+a3*x*y(x))*y(x) = 0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {4 \operatorname {a1} \left (\operatorname {a1}^{3} \operatorname {a3} \left (\operatorname {a3} \operatorname {a0} -\operatorname {a2} \sqrt {-\operatorname {a1} \operatorname {a3}}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \left (\operatorname {a2} +2\right )}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )-\frac {c_{1} \left (\operatorname {a0}^{2} \operatorname {a3} +\operatorname {a1} \,\operatorname {a2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \left (\operatorname {a2} +2\right )}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )}{4}+\operatorname {a1}^{3} \operatorname {a3} \left (\operatorname {a2} \sqrt {-\operatorname {a1} \operatorname {a3}}+\operatorname {a3} \operatorname {a0} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \operatorname {a2}}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )+\frac {\operatorname {KummerU}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \operatorname {a2}}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right ) c_{1} \left (\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} -\operatorname {a1} \operatorname {a2} \right )}{2}\right )}{4 \operatorname {a1}^{3} \operatorname {a3}^{2} \left (\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \operatorname {a2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \left (\operatorname {a2} +2\right )}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )-c_{1} \sqrt {-\operatorname {a1} \operatorname {a3}}\, \left (\operatorname {a0}^{2} \operatorname {a3} +\operatorname {a1} \,\operatorname {a2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \left (\operatorname {a2} +2\right )}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )-4 \left (\operatorname {a1}^{2} \operatorname {a3}^{2} \left (\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} -\operatorname {a1} \operatorname {a2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \operatorname {a2}}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right )-\frac {\operatorname {KummerU}\left (\frac {\sqrt {-\operatorname {a1} \operatorname {a3}}\, \operatorname {a0} +\operatorname {a1} \operatorname {a2}}{2 \operatorname {a1}}, \operatorname {a2} +1, 2 x \sqrt {-\operatorname {a1} \operatorname {a3}}\right ) c_{1} \left (\operatorname {a2} \sqrt {-\operatorname {a1} \operatorname {a3}}+\operatorname {a3} \operatorname {a0} \right )}{2}\right ) \operatorname {a1}} \]
✓ Solution by Mathematica
Time used: 0.472 (sec). Leaf size: 421
DSolve[x y'[x]+a0+a1 x+(a2+a3 x y[x])y[x]==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}}} \\ \end{align*}