Internal problem ID [11025]
Internal file name [OUTPUT/10282_Sunday_December_31_2023_03_57_16_PM_16783930/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-6 Equation of form
\((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 202.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "kovacic", "second_order_change_of_variable_on_x_method_1", "second_order_change_of_variable_on_x_method_2"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {2 y^{\prime \prime } x \left (a \,x^{2}+b x +c \right )+\left (a \,x^{2}-c \right ) y^{\prime }+\lambda \,x^{2} y=0} \]
In normal form the ode \begin {align*} \left (2 a \,x^{3}+2 b \,x^{2}+2 c x \right ) y^{\prime \prime }+\left (a \,x^{2}-c \right ) y^{\prime }+\lambda \,x^{2} y&=0 \tag {1} \end {align*}
Becomes \begin {align*} y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y&=0 \tag {2} \end {align*}
Where \begin {align*} p \left (x \right )&=\frac {a \,x^{2}-c}{2 a \,x^{3}+2 b \,x^{2}+2 c x}\\ q \left (x \right )&=\frac {x \lambda }{2 a \,x^{2}+2 b x +2 c} \end {align*}
Applying change of variables \(\tau = g \left (x \right )\) to (2) gives \begin {align*} \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+p_{1} \left (\frac {d}{d \tau }y \left (\tau \right )\right )+q_{1} y \left (\tau \right )&=0 \tag {3} \end {align*}
Where \(\tau \) is the new independent variable, and \begin {align*} p_{1} \left (\tau \right ) &=\frac {\tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {4} \\ q_{1} \left (\tau \right ) &=\frac {q \left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {5} \end {align*}
Let \(p_{1} = 0\). Eq (4) simplifies to \begin {align*} \tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )&=0 \end {align*}
This ode is solved resulting in \begin {align*} \tau &= \int {\mathrm e}^{-\left (\int p \left (x \right )d x \right )}d x\\ &= \int {\mathrm e}^{-\left (\int \frac {a \,x^{2}-c}{2 a \,x^{3}+2 b \,x^{2}+2 c x}d x \right )}d x\\ &= \int e^{\frac {\ln \left (x \right )}{2}-\frac {\ln \left (a \,x^{2}+b x +c \right )}{2}} \,dx\\ &= \int \frac {\sqrt {x}}{\sqrt {a \,x^{2}+b x +c}}d x\\ &= -\frac {\left (b +\sqrt {-4 c a +b^{2}}\right ) \sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}\, \sqrt {\frac {-2 x a +\sqrt {-4 c a +b^{2}}-b}{\sqrt {-4 c a +b^{2}}}}\, \sqrt {-\frac {a x}{b +\sqrt {-4 c a +b^{2}}}}\, \left (2 \sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 c a +b^{2}}}{\sqrt {-4 c a +b^{2}}}}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 c a +b^{2}}}{\sqrt {-4 c a +b^{2}}}}}{2}\right ) \sqrt {-4 c a +b^{2}}+\operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 c a +b^{2}}}{\sqrt {-4 c a +b^{2}}}}}{2}\right ) b \right )}{2 \sqrt {x}\, \sqrt {a \,x^{2}+b x +c}\, a^{2}}\tag {6} \end {align*}
Using (6) to evaluate \(q_{1}\) from (5) gives \begin {align*} q_{1} \left (\tau \right ) &= \frac {q \left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\\ &= \frac {\frac {x \lambda }{2 a \,x^{2}+2 b x +2 c}}{\frac {x}{a \,x^{2}+b x +c}}\\ &= \frac {\lambda }{2}\tag {7} \end {align*}
Substituting the above in (3) and noting that now \(p_{1} = 0\) results in \begin {align*} \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+q_{1} y \left (\tau \right )&=0 \\ \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+\frac {\lambda y \left (\tau \right )}{2}&=0 \end {align*}
The above ode is now solved for \(y \left (\tau \right )\).This is second order with constant coefficients homogeneous ODE. In standard form the ODE is \[ A y''(\tau ) + B y'(\tau ) + C y(\tau ) = 0 \] Where in the above \(A=1, B=0, C=\frac {\lambda }{2}\). Let the solution be \(y \left (\tau \right )=e^{\lambda \tau }\). Substituting this into the ODE gives \[ \lambda ^{2} {\mathrm e}^{\lambda \tau }+\frac {\lambda \,{\mathrm e}^{\lambda \tau }}{2} = 0 \tag {1} \] Since exponential function is never zero, then dividing Eq(2) throughout by \(e^{\lambda \tau }\) gives \[ \lambda ^{2}+\frac {\lambda }{2} = 0 \tag {2} \] Equation (2) is the characteristic equation of the ODE. Its roots determine the general solution form.Using the quadratic formula \[ \lambda _{1,2} = \frac {-B}{2 A} \pm \frac {1}{2 A} \sqrt {B^2 - 4 A C} \] Substituting \(A=1, B=0, C=\frac {\lambda }{2}\) into the above gives \begin {align*} \lambda _{1,2} &= \frac {0}{(2) \left (1\right )} \pm \frac {1}{(2) \left (1\right )} \sqrt {0^2 - (4) \left (1\right )\left (\frac {\lambda }{2}\right )}\\ &= \pm \frac {\sqrt {-2 \lambda }}{2} \end {align*}
Hence \begin{align*} \lambda _1 &= + \frac {\sqrt {-2 \lambda }}{2} \\ \lambda _2 &= - \frac {\sqrt {-2 \lambda }}{2} \\ \end{align*} Which simplifies to \begin{align*} \lambda _1 &= \frac {\sqrt {2}\, \sqrt {-\lambda }}{2} \\ \lambda _2 &= -\frac {\sqrt {2}\, \sqrt {-\lambda }}{2} \\ \end{align*} Since roots are real and distinct, then the solution is \begin{align*} y \left (\tau \right ) &= c_{1} e^{\lambda _1 \tau } + c_{2} e^{\lambda _2 \tau } \\ y \left (\tau \right ) &= c_{1} e^{\left (\frac {\sqrt {2}\, \sqrt {-\lambda }}{2}\right )\tau } +c_{2} e^{\left (-\frac {\sqrt {2}\, \sqrt {-\lambda }}{2}\right )\tau } \\ \end{align*} Or \[ y \left (\tau \right ) =c_{1} {\mathrm e}^{\frac {\sqrt {2}\, \sqrt {-\lambda }\, \tau }{2}}+c_{2} {\mathrm e}^{-\frac {\sqrt {2}\, \sqrt {-\lambda }\, \tau }{2}} \] The above solution is now transformed back to \(y\) using (6) which results in \begin {align*} y &= c_{1} {\mathrm e}^{-\frac {\sqrt {-\lambda }\, \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {\frac {-2 x a +\sqrt {-4 c a +b^{2}}-b}{\sqrt {-4 c a +b^{2}}}}\, \sqrt {-\frac {a x}{b +\sqrt {-4 c a +b^{2}}}}\, \left (b +\sqrt {-4 c a +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}}{2 \sqrt {x}\, \sqrt {a \,x^{2}+b x +c}\, a^{2}}}+c_{2} {\mathrm e}^{\frac {\sqrt {-\lambda }\, \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {\frac {-2 x a +\sqrt {-4 c a +b^{2}}-b}{\sqrt {-4 c a +b^{2}}}}\, \sqrt {-\frac {a x}{b +\sqrt {-4 c a +b^{2}}}}\, \left (b +\sqrt {-4 c a +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}}{2 \sqrt {x}\, \sqrt {a \,x^{2}+b x +c}\, a^{2}}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{-\frac {\sqrt {-\lambda }\, \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {\frac {-2 x a +\sqrt {-4 c a +b^{2}}-b}{\sqrt {-4 c a +b^{2}}}}\, \sqrt {-\frac {a x}{b +\sqrt {-4 c a +b^{2}}}}\, \left (b +\sqrt {-4 c a +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}}{2 \sqrt {x}\, \sqrt {a \,x^{2}+b x +c}\, a^{2}}}+c_{2} {\mathrm e}^{\frac {\sqrt {-\lambda }\, \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {\frac {-2 x a +\sqrt {-4 c a +b^{2}}-b}{\sqrt {-4 c a +b^{2}}}}\, \sqrt {-\frac {a x}{b +\sqrt {-4 c a +b^{2}}}}\, \left (b +\sqrt {-4 c a +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}}{2 \sqrt {x}\, \sqrt {a \,x^{2}+b x +c}\, a^{2}}} \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{-\frac {\sqrt {-\lambda }\, \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {\frac {-2 x a +\sqrt {-4 c a +b^{2}}-b}{\sqrt {-4 c a +b^{2}}}}\, \sqrt {-\frac {a x}{b +\sqrt {-4 c a +b^{2}}}}\, \left (b +\sqrt {-4 c a +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}}{2 \sqrt {x}\, \sqrt {a \,x^{2}+b x +c}\, a^{2}}}+c_{2} {\mathrm e}^{\frac {\sqrt {-\lambda }\, \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {\frac {-2 x a +\sqrt {-4 c a +b^{2}}-b}{\sqrt {-4 c a +b^{2}}}}\, \sqrt {-\frac {a x}{b +\sqrt {-4 c a +b^{2}}}}\, \left (b +\sqrt {-4 c a +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}}{2 \sqrt {x}\, \sqrt {a \,x^{2}+b x +c}\, a^{2}}} \] Verified OK.
In normal form the ode \begin {align*} \left (2 a \,x^{3}+2 b \,x^{2}+2 c x \right ) y^{\prime \prime }+\left (a \,x^{2}-c \right ) y^{\prime }+\lambda \,x^{2} y&=0 \tag {1} \end {align*}
Becomes \begin {align*} y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y&=0 \tag {2} \end {align*}
Where \begin {align*} p \left (x \right )&=\frac {a \,x^{2}-c}{2 x \left (a \,x^{2}+b x +c \right )}\\ q \left (x \right )&=\frac {x \lambda }{2 a \,x^{2}+2 b x +2 c} \end {align*}
Applying change of variables \(\tau = g \left (x \right )\) to (2) results \begin {align*} \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+p_{1} \left (\frac {d}{d \tau }y \left (\tau \right )\right )+q_{1} y \left (\tau \right )&=0 \tag {3} \end {align*}
Where \(\tau \) is the new independent variable, and \begin {align*} p_{1} \left (\tau \right ) &=\frac {\tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {4} \\ q_{1} \left (\tau \right ) &=\frac {q \left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {5} \end {align*}
Let \(q_1=c^2\) where \(c\) is some constant. Therefore from (5) \begin {align*} \tau ' &= \frac {1}{c}\sqrt {q}\\ &= \frac {\sqrt {\frac {x \lambda }{2 a \,x^{2}+2 b x +2 c}}}{c}\tag {6} \\ \tau '' &= \frac {\frac {\lambda }{2 a \,x^{2}+2 b x +2 c}-\frac {x \lambda \left (4 x a +2 b \right )}{\left (2 a \,x^{2}+2 b x +2 c \right )^{2}}}{2 c \sqrt {\frac {x \lambda }{2 a \,x^{2}+2 b x +2 c}}} \end {align*}
Substituting the above into (4) results in \begin {align*} p_{1} \left (\tau \right ) &=\frac {\tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\\ &=\frac {\frac {\frac {\lambda }{2 a \,x^{2}+2 b x +2 c}-\frac {x \lambda \left (4 x a +2 b \right )}{\left (2 a \,x^{2}+2 b x +2 c \right )^{2}}}{2 c \sqrt {\frac {x \lambda }{2 a \,x^{2}+2 b x +2 c}}}+\frac {a \,x^{2}-c}{2 x \left (a \,x^{2}+b x +c \right )}\frac {\sqrt {\frac {x \lambda }{2 a \,x^{2}+2 b x +2 c}}}{c}}{\left (\frac {\sqrt {\frac {x \lambda }{2 a \,x^{2}+2 b x +2 c}}}{c}\right )^2} \\ &=0 \end {align*}
Therefore ode (3) now becomes \begin {align*} y \left (\tau \right )'' + p_1 y \left (\tau \right )' + q_1 y \left (\tau \right ) &= 0 \\ \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+c^{2} y \left (\tau \right ) &= 0 \tag {7} \end {align*}
The above ode is now solved for \(y \left (\tau \right )\). Since the ode is now constant coefficients, it can be easily solved to give \begin {align*} y \left (\tau \right ) &= c_{1} \cos \left (c \tau \right )+c_{2} \sin \left (c \tau \right ) \end {align*}
Now from (6) \begin {align*} \tau &= \int \frac {1}{c} \sqrt q \,dx \\ &= \frac {\int \sqrt {\frac {x \lambda }{2 a \,x^{2}+2 b x +2 c}}d x}{c}\\ &= -\frac {\sqrt {\frac {x \lambda }{a \,x^{2}+b x +c}}\, \left (a \,x^{2}+b x +c \right ) \left (b +\sqrt {-4 c a +b^{2}}\right ) \sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}\, \sqrt {\frac {-2 x a +\sqrt {-4 c a +b^{2}}-b}{\sqrt {-4 c a +b^{2}}}}\, \sqrt {-\frac {2 a x}{b +\sqrt {-4 c a +b^{2}}}}\, \left (2 \sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 c a +b^{2}}}{\sqrt {-4 c a +b^{2}}}}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 c a +b^{2}}}{\sqrt {-4 c a +b^{2}}}}}{2}\right ) \sqrt {-4 c a +b^{2}}+\operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-4 c a +b^{2}}}{\sqrt {-4 c a +b^{2}}}}}{2}\right ) b \right )}{4 c \sqrt {x \left (a \,x^{2}+b x +c \right )}\, a^{2} \sqrt {a \,x^{3}+b \,x^{2}+c x}} \end {align*}
Substituting the above into the solution obtained gives \[ y = c_{1} \cos \left (\frac {\sqrt {\lambda }\, \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {\frac {-2 x a +\sqrt {-4 c a +b^{2}}-b}{\sqrt {-4 c a +b^{2}}}}\, \sqrt {-\frac {1}{b +\sqrt {-4 c a +b^{2}}}}\, \left (b +\sqrt {-4 c a +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}}{2 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x +c}}\right )-c_{2} \sin \left (\frac {\sqrt {\lambda }\, \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {\frac {-2 x a +\sqrt {-4 c a +b^{2}}-b}{\sqrt {-4 c a +b^{2}}}}\, \sqrt {-\frac {1}{b +\sqrt {-4 c a +b^{2}}}}\, \left (b +\sqrt {-4 c a +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}}{2 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x +c}}\right ) \]
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \cos \left (\frac {\sqrt {\lambda }\, \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {\frac {-2 x a +\sqrt {-4 c a +b^{2}}-b}{\sqrt {-4 c a +b^{2}}}}\, \sqrt {-\frac {1}{b +\sqrt {-4 c a +b^{2}}}}\, \left (b +\sqrt {-4 c a +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}}{2 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x +c}}\right )-c_{2} \sin \left (\frac {\sqrt {\lambda }\, \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {\frac {-2 x a +\sqrt {-4 c a +b^{2}}-b}{\sqrt {-4 c a +b^{2}}}}\, \sqrt {-\frac {1}{b +\sqrt {-4 c a +b^{2}}}}\, \left (b +\sqrt {-4 c a +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}}{2 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x +c}}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \cos \left (\frac {\sqrt {\lambda }\, \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {\frac {-2 x a +\sqrt {-4 c a +b^{2}}-b}{\sqrt {-4 c a +b^{2}}}}\, \sqrt {-\frac {1}{b +\sqrt {-4 c a +b^{2}}}}\, \left (b +\sqrt {-4 c a +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}}{2 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x +c}}\right )-c_{2} \sin \left (\frac {\sqrt {\lambda }\, \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {\frac {-2 x a +\sqrt {-4 c a +b^{2}}-b}{\sqrt {-4 c a +b^{2}}}}\, \sqrt {-\frac {1}{b +\sqrt {-4 c a +b^{2}}}}\, \left (b +\sqrt {-4 c a +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}}{2 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x +c}}\right ) \] Warning, solution could not be verified
Writing the ode as \begin {align*} \left (2 a \,x^{3}+2 b \,x^{2}+2 c x \right ) y^{\prime \prime }+\left (a \,x^{2}-c \right ) y^{\prime }+\lambda \,x^{2} y &= 0 \tag {1} \\ A y^{\prime \prime } + B y^{\prime } + C y &= 0 \tag {2} \end {align*}
Comparing (1) and (2) shows that \begin {align*} A &= 2 a \,x^{3}+2 b \,x^{2}+2 c x \\ B &= a \,x^{2}-c\tag {3} \\ C &= x^{2} \lambda \end {align*}
Applying the Liouville transformation on the dependent variable gives \begin {align*} z(x) &= y e^{\int \frac {B}{2 A} \,dx} \end {align*}
Then (2) becomes \begin {align*} z''(x) = r z(x)\tag {4} \end {align*}
Where \(r\) is given by \begin {align*} r &= \frac {s}{t}\tag {5} \\ &= \frac {2 A B' - 2 B A' + B^2 - 4 A C}{4 A^2} \end {align*}
Substituting the values of \(A,B,C\) from (3) in the above and simplifying gives \begin {align*} r &= \frac {-8 a \lambda \,x^{5}-3 x^{4} a^{2}-8 b \lambda \,x^{4}-8 c \lambda \,x^{3}+14 a c \,x^{2}+8 b c x +5 c^{2}}{16 \left (a \,x^{3}+b \,x^{2}+c x \right )^{2}}\tag {6} \end {align*}
Comparing the above to (5) shows that \begin {align*} s &= -8 a \lambda \,x^{5}-3 x^{4} a^{2}-8 b \lambda \,x^{4}-8 c \lambda \,x^{3}+14 a c \,x^{2}+8 b c x +5 c^{2}\\ t &= 16 \left (a \,x^{3}+b \,x^{2}+c x \right )^{2} \end {align*}
Therefore eq. (4) becomes \begin {align*} z''(x) &= \left ( \frac {-8 a \lambda \,x^{5}-3 x^{4} a^{2}-8 b \lambda \,x^{4}-8 c \lambda \,x^{3}+14 a c \,x^{2}+8 b c x +5 c^{2}}{16 \left (a \,x^{3}+b \,x^{2}+c x \right )^{2}}\right ) z(x)\tag {7} \end {align*}
Equation (7) is now solved. After finding \(z(x)\) then \(y\) is found using the inverse transformation \begin {align*} y &= z \left (x \right ) e^{-\int \frac {B}{2 A} \,dx} \end {align*}
The first step is to determine the case of Kovacic algorithm this ode belongs to. There are 3 cases depending on the order of poles of \(r\) and the order of \(r\) at \(\infty \). The following table summarizes these cases.
| Case |
Allowed pole order for \(r\) |
Allowed value for \(\mathcal {O}(\infty )\) |
| 1 |
\(\left \{ 0,1,2,4,6,8,\cdots \right \} \) |
\(\left \{ \cdots ,-6,-4,-2,0,2,3,4,5,6,\cdots \right \} \) |
|
2 |
Need to have at least one pole that is either order \(2\) or odd order greater than \(2\). Any other pole order is allowed as long as the above condition is satisfied. Hence the following set of pole orders are all allowed. \(\{1,2\}\),\(\{1,3\}\),\(\{2\}\),\(\{3\}\),\(\{3,4\}\),\(\{1,2,5\}\). |
no condition |
| 3 |
\(\left \{ 1,2\right \} \) |
\(\left \{ 2,3,4,5,6,7,\cdots \right \} \) |
The order of \(r\) at \(\infty \) is the degree of \(t\) minus the degree of \(s\). Therefore \begin {align*} O\left (\infty \right ) &= \text {deg}(t) - \text {deg}(s) \\ &= 6 - 5 \\ &= 1 \end {align*}
The poles of \(r\) in eq. (7) and the order of each pole are determined by solving for the roots of \(t=16 \left (a \,x^{3}+b \,x^{2}+c x \right )^{2}\). There is a pole at \(x=0\) of order \(2\). There is a pole at \(x=-\frac {b -\sqrt {-4 c a +b^{2}}}{2 a}\) of order \(2\). There is a pole at \(x=-\frac {b +\sqrt {-4 c a +b^{2}}}{2 a}\) of order \(2\). Since there is a pole of order \(2\) then necessary conditions for case two are met. Therefore \begin {align*} L &= [2] \end {align*}
Attempting to find a solution using case \(n=2\).
Looking at poles of order 2. The partial fractions decomposition of \(r\) is \[ r = \frac {5}{16 x^{2}}-\frac {b}{8 c x}+\frac {\frac {\left (-b +\sqrt {-4 c a +b^{2}}\right )^{3} b}{4 a}-\frac {\left (-b +\sqrt {-4 c a +b^{2}}\right )^{3} c \lambda }{a^{2}}-2 \left (-b +\sqrt {-4 c a +b^{2}}\right )^{2} c +\frac {\left (-b +\sqrt {-4 c a +b^{2}}\right )^{2} b^{2}}{a}-\frac {2 \left (-b +\sqrt {-4 c a +b^{2}}\right )^{2} b c \lambda }{a^{2}}-3 \left (-b +\sqrt {-4 c a +b^{2}}\right ) b c +\frac {\left (-b +\sqrt {-4 c a +b^{2}}\right ) b^{3}}{a}-\frac {4 \left (-b +\sqrt {-4 c a +b^{2}}\right ) c^{2} \lambda }{a}+4 a \,c^{2}-b^{2} c}{16 c \left (-4 c a +b^{2}\right ) {\left (x -\frac {-b +\sqrt {-4 c a +b^{2}}}{2 a}\right )}^{2}}+\frac {-\frac {\left (b +\sqrt {-4 c a +b^{2}}\right )^{3} b}{4 a}+\frac {\left (b +\sqrt {-4 c a +b^{2}}\right )^{3} c \lambda }{a^{2}}-2 \left (b +\sqrt {-4 c a +b^{2}}\right )^{2} c +\frac {\left (b +\sqrt {-4 c a +b^{2}}\right )^{2} b^{2}}{a}-\frac {2 \left (b +\sqrt {-4 c a +b^{2}}\right )^{2} b c \lambda }{a^{2}}+3 \left (b +\sqrt {-4 c a +b^{2}}\right ) b c -\frac {\left (b +\sqrt {-4 c a +b^{2}}\right ) b^{3}}{a}+\frac {4 \left (b +\sqrt {-4 c a +b^{2}}\right ) c^{2} \lambda }{a}+4 a \,c^{2}-b^{2} c}{16 c \left (-4 c a +b^{2}\right ) {\left (x +\frac {b +\sqrt {-4 c a +b^{2}}}{2 a}\right )}^{2}}+\frac {\frac {\left (-b +\sqrt {-4 c a +b^{2}}\right )^{3} b}{2}-\frac {2 \left (-b +\sqrt {-4 c a +b^{2}}\right )^{3} c \lambda }{a}-2 a \left (-b +\sqrt {-4 c a +b^{2}}\right )^{2} c +\frac {7 \left (-b +\sqrt {-4 c a +b^{2}}\right )^{2} b^{2}}{4}-\frac {5 \left (-b +\sqrt {-4 c a +b^{2}}\right )^{2} b c \lambda }{a}-4 a \left (-b +\sqrt {-4 c a +b^{2}}\right ) b c +2 \left (-b +\sqrt {-4 c a +b^{2}}\right ) b^{3}-\frac {4 \left (-b +\sqrt {-4 c a +b^{2}}\right ) b^{2} c \lambda }{a}-4 a^{2} c^{2}-2 a \,b^{2} c +b^{4}-4 b \,c^{2} \lambda }{8 c \left (-4 c a +b^{2}\right )^{\frac {3}{2}} \left (x -\frac {-b +\sqrt {-4 c a +b^{2}}}{2 a}\right )}-\frac {-\frac {\left (b +\sqrt {-4 c a +b^{2}}\right )^{3} b}{2}+\frac {2 \left (b +\sqrt {-4 c a +b^{2}}\right )^{3} c \lambda }{a}-2 \left (b +\sqrt {-4 c a +b^{2}}\right )^{2} a c +\frac {7 \left (b +\sqrt {-4 c a +b^{2}}\right )^{2} b^{2}}{4}-\frac {5 \left (b +\sqrt {-4 c a +b^{2}}\right )^{2} b c \lambda }{a}+4 \left (b +\sqrt {-4 c a +b^{2}}\right ) a b c -2 \left (b +\sqrt {-4 c a +b^{2}}\right ) b^{3}+\frac {4 \left (b +\sqrt {-4 c a +b^{2}}\right ) b^{2} c \lambda }{a}-4 a^{2} c^{2}-2 a \,b^{2} c +b^{4}-4 b \,c^{2} \lambda }{8 c \left (-4 c a +b^{2}\right )^{\frac {3}{2}} \left (x +\frac {b +\sqrt {-4 c a +b^{2}}}{2 a}\right )} \] For the pole at \(x=0\) let \(b\) be the coefficient of \(\frac {1}{ x^{2}}\) in the partial fractions decomposition of \(r\) given above. Therefore \(b={\frac {5}{16}}\). Hence \begin {align*} E_c &= \{2, 2+2\sqrt {1+4 b}, 2-2\sqrt {1+4 b} \} \\ &= \{-1, 2, 5\} \end {align*}
For the pole at \(x=-\frac {b -\sqrt {-4 c a +b^{2}}}{2 a}\) let \(b\) be the coefficient of \(\frac {1}{ {\left (x +\frac {b -\sqrt {-4 c a +b^{2}}}{2 a}\right )}^{2}}\) in the partial fractions decomposition of \(r\) given above. Therefore \(b=-{\frac {3}{16}}\). Hence \begin {align*} E_c &= \{2, 2+2\sqrt {1+4 b}, 2-2\sqrt {1+4 b} \} \\ &= \{1, 2, 3\} \end {align*}
For the pole at \(x=-\frac {b +\sqrt {-4 c a +b^{2}}}{2 a}\) let \(b\) be the coefficient of \(\frac {1}{ {\left (x +\frac {b +\sqrt {-4 c a +b^{2}}}{2 a}\right )}^{2}}\) in the partial fractions decomposition of \(r\) given above. Therefore \(b=-{\frac {3}{16}}\). Hence \begin {align*} E_c &= \{2, 2+2\sqrt {1+4 b}, 2-2\sqrt {1+4 b} \} \\ &= \{1, 2, 3\} \end {align*}
Since the order of \(r\) at \(\infty \) is \(1 <2\) then \begin {align*} E_\infty = \{1\} \end {align*}
The following table summarizes the findings so far for poles and for the order of \(r\) at \(\infty \) for case 2 of Kovacic algorithm.
| pole \(c\) location | pole order | \(E_c\) |
| \(0\) | \(2\) | \(\{-1, 2, 5\}\) |
| \(-\frac {b -\sqrt {-4 c a +b^{2}}}{2 a}\) | \(2\) | \(\{1, 2, 3\}\) |
| \(-\frac {b +\sqrt {-4 c a +b^{2}}}{2 a}\) | \(2\) | \(\{1, 2, 3\}\) |
| Order of \(r\) at \(\infty \) | \(E_\infty \) |
| \(1\) | \(\{1\}\) |
Using the family \(\{e_1,e_2,\dots ,e_\infty \}\) given by \[ e_1=-1,\hspace {3pt} e_2=1,\hspace {3pt} e_3=1,\hspace {3pt} e_\infty =1 \] Gives a non negative integer \(d\) (the degree of the polynomial \(p(x)\)), which is generated using \begin {align*} d &= \frac {1}{2} \left ( e_\infty - \sum _{c \in \Gamma } e_c \right )\\ &= \frac {1}{2} \left ( 1 - \left (-1+\left (1\right )+\left (1\right )\right )\right )\\ &= 0 \end {align*}
We now form the following rational function \begin {align*} \theta &= \frac {1}{2} \sum _{c \in \Gamma } \frac {e_c}{x-c} \\ &= \frac {1}{2} \left (\frac {-1}{\left (x-\left (0\right )\right )}+\frac {1}{\left (x-\left (-\frac {b -\sqrt {-4 c a +b^{2}}}{2 a}\right )\right )}+\frac {1}{\left (x-\left (-\frac {b +\sqrt {-4 c a +b^{2}}}{2 a}\right )\right )}\right ) \\ &= -\frac {1}{2 x}+\frac {1}{2 x +\frac {b -\sqrt {-4 c a +b^{2}}}{a}}+\frac {1}{2 x +\frac {b +\sqrt {-4 c a +b^{2}}}{a}} \end {align*}
Now we search for a monic polynomial \(p(x)\) of degree \(d=0\) such that \[ p'''+3 \theta p'' + \left (3 \theta ^2 + 3 \theta ' - 4 r\right )p' + \left (\theta '' + 3 \theta \theta ' + \theta ^3 - 4 r \theta - 2 r' \right ) p = 0 \tag {1A} \] Since \(d=0\), then letting \[ p = 1\tag {2A} \] Substituting \(p\) and \(\theta \) into Eq. (1A) gives \[ 0 = 0 \] And solving for \(p\) gives \[ p = 1 \] Now that \(p(x)\) is found let \begin {align*} \phi &= \theta + \frac {p'}{p}\\ &= -\frac {1}{2 x}+\frac {1}{2 x +\frac {b -\sqrt {-4 c a +b^{2}}}{a}}+\frac {1}{2 x +\frac {b +\sqrt {-4 c a +b^{2}}}{a}} \end {align*}
Let \(\omega \) be the solution of \begin {align*} \omega ^2 - \phi \omega + \left ( \frac {1}{2} \phi ' + \frac {1}{2} \phi ^2 - r \right ) &= 0 \end {align*}
Substituting the values for \(\phi \) and \(r\) into the above equation gives \[ w^{2}-\left (-\frac {1}{2 x}+\frac {1}{2 x +\frac {b -\sqrt {-4 c a +b^{2}}}{a}}+\frac {1}{2 x +\frac {b +\sqrt {-4 c a +b^{2}}}{a}}\right ) w +\frac {a^{2} \left (8 a \lambda \,x^{5}+\left (a^{2}+8 b \lambda \right ) x^{4}+8 c \lambda \,x^{3}-2 a c \,x^{2}+c^{2}\right )}{x^{2} \left (2 x a +b -\sqrt {-4 c a +b^{2}}\right )^{2} \left (2 x a +\sqrt {-4 c a +b^{2}}+b \right )^{2}} = 0 \] Solving for \(\omega \) gives \begin {align*} \omega &= \frac {a \,x^{2}+2 x \sqrt {2}\, \sqrt {-x \lambda \left (a \,x^{2}+b x +c \right )}-c}{4 x \left (a \,x^{2}+b x +c \right )} \end {align*}
Therefore the first solution to the ode \(z'' = r z\) is \begin {align*} z_1(x) &= e^{ \int \omega \,dx} \\ &= {\mathrm e}^{\int \frac {a \,x^{2}+2 x \sqrt {2}\, \sqrt {-x \lambda \left (a \,x^{2}+b x +c \right )}-c}{4 x \left (a \,x^{2}+b x +c \right )}d x}\\ &= \frac {\left (a \,x^{2}+b x +c \right )^{\frac {1}{4}} {\mathrm e}^{\frac {\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}\, \lambda \sqrt {\frac {-2 x a +\sqrt {-4 c a +b^{2}}-b}{\sqrt {-4 c a +b^{2}}}}\, \sqrt {2}\, \sqrt {-\frac {a x}{b +\sqrt {-4 c a +b^{2}}}}\, \left (b +\sqrt {-4 c a +b^{2}}\right ) \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {\sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{2 \left (-4 c a +b^{2}\right )^{\frac {1}{4}}}\right )\right )}{2 \sqrt {-x \lambda \left (a \,x^{2}+b x +c \right )}\, a^{2}}}}{x^{\frac {1}{4}}} \end {align*}
The first solution to the original ode in \(y\) is found from \begin{align*} y_1 &= z_1 e^{ \int -\frac {1}{2} \frac {B}{A} \,dx} \\ &= z_1 e^{ -\int \frac {1}{2} \frac {a \,x^{2}-c}{2 a \,x^{3}+2 b \,x^{2}+2 c x} \,dx} \\ &= z_1 e^{\frac {\ln \left (x \right )}{4}-\frac {\ln \left (a \,x^{2}+b x +c \right )}{4}} \\ &= z_1 \left (\frac {x^{\frac {1}{4}}}{\left (a \,x^{2}+b x +c \right )^{\frac {1}{4}}}\right ) \\ \end{align*} Which simplifies to \[ y_1 = {\mathrm e}^{\frac {\lambda \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )\right ) \sqrt {-\frac {a^{2}}{\lambda \sqrt {-4 c a +b^{2}}}}\, \sqrt {2}}{a^{2}}} \] The second solution \(y_2\) to the original ode is found using reduction of order \[ y_2 = y_1 \int \frac { e^{\int -\frac {B}{A} \,dx}}{y_1^2} \,dx \] Substituting gives \begin{align*} y_2 &= y_1 \int \frac { e^{\int -\frac {a \,x^{2}-c}{2 a \,x^{3}+2 b \,x^{2}+2 c x} \,dx}}{\left (y_1\right )^2} \,dx \\ &= y_1 \int \frac { e^{\frac {\ln \left (x \right )}{2}-\frac {\ln \left (a \,x^{2}+b x +c \right )}{2}}}{\left (y_1\right )^2} \,dx \\ &= y_1 \left (\int \frac {\sqrt {x}\, {\mathrm e}^{-\frac {2 \lambda \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )\right ) \sqrt {-\frac {a^{2}}{\lambda \sqrt {-4 c a +b^{2}}}}\, \sqrt {2}}{a^{2}}}}{\sqrt {a \,x^{2}+b x +c}}d x\right ) \\ \end{align*} Therefore the solution is
\begin{align*} y &= c_{1} y_1 + c_{2} y_2 \\ &= c_{1} \left ({\mathrm e}^{\frac {\lambda \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )\right ) \sqrt {-\frac {a^{2}}{\lambda \sqrt {-4 c a +b^{2}}}}\, \sqrt {2}}{a^{2}}}\right ) + c_{2} \left ({\mathrm e}^{\frac {\lambda \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )\right ) \sqrt {-\frac {a^{2}}{\lambda \sqrt {-4 c a +b^{2}}}}\, \sqrt {2}}{a^{2}}}\left (\int \frac {\sqrt {x}\, {\mathrm e}^{-\frac {2 \lambda \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )\right ) \sqrt {-\frac {a^{2}}{\lambda \sqrt {-4 c a +b^{2}}}}\, \sqrt {2}}{a^{2}}}}{\sqrt {a \,x^{2}+b x +c}}d x\right )\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{\frac {\lambda \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )\right ) \sqrt {-\frac {a^{2}}{\lambda \sqrt {-4 c a +b^{2}}}}\, \sqrt {2}}{a^{2}}}+c_{2} {\mathrm e}^{\frac {\lambda \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )\right ) \sqrt {-\frac {a^{2}}{\lambda \sqrt {-4 c a +b^{2}}}}\, \sqrt {2}}{a^{2}}} \left (\int \frac {\sqrt {x}\, {\mathrm e}^{-\frac {2 \lambda \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )\right ) \sqrt {-\frac {a^{2}}{\lambda \sqrt {-4 c a +b^{2}}}}\, \sqrt {2}}{a^{2}}}}{\sqrt {a \,x^{2}+b x +c}}d x \right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} {\mathrm e}^{\frac {\lambda \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )\right ) \sqrt {-\frac {a^{2}}{\lambda \sqrt {-4 c a +b^{2}}}}\, \sqrt {2}}{a^{2}}}+c_{2} {\mathrm e}^{\frac {\lambda \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )\right ) \sqrt {-\frac {a^{2}}{\lambda \sqrt {-4 c a +b^{2}}}}\, \sqrt {2}}{a^{2}}} \left (\int \frac {\sqrt {x}\, {\mathrm e}^{-\frac {2 \lambda \left (\frac {\left (b -\sqrt {-4 c a +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )}{2}+\sqrt {-4 c a +b^{2}}\, \operatorname {EllipticE}\left (\sqrt {\frac {2 x a +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}}, \frac {{\left (-\left (4 c a -b^{2}\right )^{3}\right )}^{\frac {1}{4}} \sqrt {2 b +2 \sqrt {-4 c a +b^{2}}}}{8 c a -2 b^{2}}\right )\right ) \sqrt {-\frac {a^{2}}{\lambda \sqrt {-4 c a +b^{2}}}}\, \sqrt {2}}{a^{2}}}}{\sqrt {a \,x^{2}+b x +c}}d x \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (2 a \,x^{3}+2 b \,x^{2}+2 c x \right ) \left (\frac {d}{d x}y^{\prime }\right )+\left (a \,x^{2}-c \right ) y^{\prime }+\lambda \,x^{2} y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=-\frac {\lambda x y}{2 \left (a \,x^{2}+b x +c \right )}-\frac {\left (a \,x^{2}-c \right ) y^{\prime }}{2 x \left (a \,x^{2}+b x +c \right )} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }+\frac {\left (a \,x^{2}-c \right ) y^{\prime }}{2 x \left (a \,x^{2}+b x +c \right )}+\frac {\lambda x y}{2 \left (a \,x^{2}+b x +c \right )}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {a \,x^{2}-c}{2 x \left (a \,x^{2}+b x +c \right )}, P_{3}\left (x \right )=\frac {x \lambda }{2 \left (a \,x^{2}+b x +c \right )}\right ] \\ {} & \circ & x \cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x \cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=-\frac {1}{2} \\ {} & \circ & x^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=0 \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & 2 \left (\frac {d}{d x}y^{\prime }\right ) x \left (a \,x^{2}+b x +c \right )+\left (a \,x^{2}-c \right ) y^{\prime }+\lambda \,x^{2} y=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{2}\cdot y\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r +2} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -2 \\ {} & {} & x^{2}\cdot y=\moverset {\infty }{\munderset {k =2}{\sum }}a_{k -2} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..2 \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..3 \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r -2+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2-m \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =-2+m}{\sum }}a_{k +2-m} \left (k +2-m +r \right ) \left (k +1-m +r \right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & c a_{0} r \left (-3+2 r \right ) x^{-1+r}+\left (c a_{1} \left (1+r \right ) \left (-1+2 r \right )+2 a_{0} r \left (-1+r \right ) b \right ) x^{r}+\left (c a_{2} \left (2+r \right ) \left (1+2 r \right )+2 a_{1} \left (1+r \right ) r b +a a_{0} r \left (-1+2 r \right )\right ) x^{1+r}+\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (c a_{k +1} \left (k +1+r \right ) \left (2 k +2 r -1\right )+2 a_{k} \left (k +r \right ) \left (k +r -1\right ) b +a a_{k -1} \left (k +r -1\right ) \left (2 k +2 r -3\right )+\lambda a_{k -2}\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & c r \left (-3+2 r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, \frac {3}{2}\right \} \\ \bullet & {} & \textrm {The coefficients of each power of}\hspace {3pt} x \hspace {3pt}\textrm {must be 0}\hspace {3pt} \\ {} & {} & \left [c a_{1} \left (1+r \right ) \left (-1+2 r \right )+2 a_{0} r \left (-1+r \right ) b =0, c a_{2} \left (2+r \right ) \left (1+2 r \right )+2 a_{1} \left (1+r \right ) r b +a a_{0} r \left (-1+2 r \right )=0\right ] \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & \left \{a_{1}=-\frac {2 a_{0} r \left (-1+r \right ) b}{c \left (2 r^{2}+r -1\right )}, a_{2}=-\frac {a_{0} r \left (4 a c \,r^{2}-4 b^{2} r^{2}-4 a c r +4 b^{2} r +c a \right )}{c^{2} \left (4 r^{3}+8 r^{2}-r -2\right )}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & c a_{k +1} \left (k +1+r \right ) \left (2 k +2 r -1\right )+2 a_{k} \left (k +r \right ) \left (k +r -1\right ) b +a a_{k -1} \left (k +r -1\right ) \left (2 k +2 r -3\right )+\lambda a_{k -2}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & c a_{k +3} \left (k +3+r \right ) \left (2 k +3+2 r \right )+2 a_{k +2} \left (k +r +2\right ) \left (k +1+r \right ) b +a a_{k +1} \left (k +1+r \right ) \left (2 k +1+2 r \right )+\lambda a_{k}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +3}=-\frac {2 a \,k^{2} a_{k +1}+4 a k r a_{k +1}+2 a \,r^{2} a_{k +1}+2 b \,k^{2} a_{k +2}+4 b k r a_{k +2}+2 b \,r^{2} a_{k +2}+3 a k a_{k +1}+3 a r a_{k +1}+6 b k a_{k +2}+6 b r a_{k +2}+a a_{k +1}+4 b a_{k +2}+\lambda a_{k}}{c \left (k +3+r \right ) \left (2 k +3+2 r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +3}=-\frac {2 a \,k^{2} a_{k +1}+2 b \,k^{2} a_{k +2}+3 a k a_{k +1}+6 b k a_{k +2}+a a_{k +1}+4 b a_{k +2}+\lambda a_{k}}{c \left (k +3\right ) \left (2 k +3\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =0 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}, a_{k +3}=-\frac {2 a \,k^{2} a_{k +1}+2 b \,k^{2} a_{k +2}+3 a k a_{k +1}+6 b k a_{k +2}+a a_{k +1}+4 b a_{k +2}+\lambda a_{k}}{c \left (k +3\right ) \left (2 k +3\right )}, a_{1}=0, a_{2}=0\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {3}{2} \\ {} & {} & a_{k +3}=-\frac {2 a \,k^{2} a_{k +1}+2 b \,k^{2} a_{k +2}+9 a k a_{k +1}+12 b k a_{k +2}+10 a a_{k +1}+\frac {35}{2} b a_{k +2}+\lambda a_{k}}{c \left (k +\frac {9}{2}\right ) \left (2 k +6\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {3}{2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {3}{2}}, a_{k +3}=-\frac {2 a \,k^{2} a_{k +1}+2 b \,k^{2} a_{k +2}+9 a k a_{k +1}+12 b k a_{k +2}+10 a a_{k +1}+\frac {35}{2} b a_{k +2}+\lambda a_{k}}{c \left (k +\frac {9}{2}\right ) \left (2 k +6\right )}, a_{1}=-\frac {3 a_{0} b}{10 c}, a_{2}=-\frac {3 a_{0} \left (4 c a -3 b^{2}\right )}{56 c^{2}}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}d_{k} x^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}e_{k} x^{k +\frac {3}{2}}\right ), d_{k +3}=-\frac {2 a \,k^{2} d_{k +1}+2 b \,k^{2} d_{k +2}+3 a k d_{k +1}+6 b k d_{k +2}+a d_{k +1}+4 b d_{k +2}+\lambda d_{k}}{c \left (k +3\right ) \left (2 k +3\right )}, d_{1}=0, d_{2}=0, e_{k +3}=-\frac {2 a \,k^{2} e_{k +1}+2 b \,k^{2} e_{k +2}+9 a k e_{k +1}+12 b k e_{k +2}+10 a e_{k +1}+\frac {35}{2} b e_{k +2}+\lambda e_{k}}{c \left (k +\frac {9}{2}\right ) \left (2 k +6\right )}, e_{1}=-\frac {3 e_{0} b}{10 c}, e_{2}=-\frac {3 e_{0} \left (4 c a -3 b^{2}\right )}{56 c^{2}}\right ] \end {array} \]
Maple trace Kovacic algorithm successful
`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)] Solution is available but has integrals. Trying a simpler solution using Kovacics algorithm... -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Group is reducible or imprimitive Solution has integrals. Trying a special function solution free of integrals... -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius -> Mathieu -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius No special function solution was found. <- Kovacics algorithm successful Solution via Kovacic is not simpler. Returning default solution <- linear_1 successful`
✓ Solution by Maple
Time used: 7.125 (sec). Leaf size: 65
dsolve(2*x*(a*x^2+b*x+c)*diff(y(x),x$2)+(a*x^2-c)*diff(y(x),x)+lambda*x^2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {i \sqrt {2}\, \sqrt {\lambda }\, \left (\int \frac {\sqrt {x}}{\sqrt {a \,x^{2}+b x +c}}d x \right )}{2}}+c_{2} {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {\lambda }\, \left (\int \frac {\sqrt {x}}{\sqrt {a \,x^{2}+b x +c}}d x \right )}{2}} \]
✓ Solution by Mathematica
Time used: 144.69 (sec). Leaf size: 501
DSolve[2*x*(a*x^2+b*x+c)*y''[x]+(a*x^2-c)*y'[x]+\[Lambda]*x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \cosh \left (\frac {\sqrt {\lambda } \left (\sqrt {b^2-4 a c}-b\right ) \sqrt {\sqrt {b^2-4 a c}+2 a x+b} \sqrt {\frac {2 a x}{b-\sqrt {b^2-4 a c}}+1} \left (E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{2 a^{3/2} \sqrt {x (a x+b)+c}}\right )+i c_2 \sinh \left (\frac {\sqrt {\lambda } \left (\sqrt {b^2-4 a c}-b\right ) \sqrt {\sqrt {b^2-4 a c}+2 a x+b} \sqrt {\frac {2 a x}{b-\sqrt {b^2-4 a c}}+1} \left (E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{2 a^{3/2} \sqrt {x (a x+b)+c}}\right ) \]