Internal problem ID [10960]
Internal file name [OUTPUT/10217_Sunday_December_31_2023_11_10_08_AM_81546286/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 2.1.2-4 Equation of form
\(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 137.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }-b y=0} \]
In normal form the ode \begin {align*} x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime } x -b y = 0 \tag {1} \end {align*}
Becomes \begin {align*} y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y&=r \left (x \right ) \tag {2} \end {align*}
Where \begin {align*} p \left (x \right )&=\frac {a x +b}{x}\\ q \left (x \right )&=-\frac {b}{x^{2}}\\ r \left (x \right )&=0 \end {align*}
The Lagrange adjoint ode is given by \begin {align*} \xi ^{''}-(\xi \, p)'+\xi q &= 0\\ \xi ^{''}-\left (\frac {\left (a x +b \right ) \xi \left (x \right )}{x}\right )' + \left (-\frac {b \xi \left (x \right )}{x^{2}}\right ) &= 0\\ \xi ^{\prime \prime }\left (x \right )-\frac {\left (a x +b \right ) \xi ^{\prime }\left (x \right )}{x}+\left (-\frac {a}{x}+\frac {a x +b}{x^{2}}-\frac {b}{x^{2}}\right ) \xi \left (x \right )&= 0 \end {align*}
Which is solved for \(\xi (x)\). This is second order ode with missing dependent variable \(\xi \left (x \right )\). Let \begin {align*} p(x) &= \xi ^{\prime }\left (x \right ) \end {align*}
Then \begin {align*} p'(x) &= \xi ^{\prime \prime }\left (x \right ) \end {align*}
Hence the ode becomes \begin {align*} p^{\prime }\left (x \right ) x +\left (-a x -b \right ) p \left (x \right ) = 0 \end {align*}
Which is now solve for \(p(x)\) as first order ode. In canonical form the ODE is \begin {align*} p' &= F(x,p)\\ &= f( x) g(p)\\ &= \frac {p \left (a x +b \right )}{x} \end {align*}
Where \(f(x)=\frac {a x +b}{x}\) and \(g(p)=p\). Integrating both sides gives \begin {align*} \frac {1}{p} \,dp &= \frac {a x +b}{x} \,d x\\ \int { \frac {1}{p} \,dp} &= \int {\frac {a x +b}{x} \,d x}\\ \ln \left (p \right )&=a x +b \ln \left (x \right )+c_{1}\\ p&={\mathrm e}^{a x +b \ln \left (x \right )+c_{1}}\\ &=c_{1} {\mathrm e}^{a x +b \ln \left (x \right )} \end {align*}
Which simplifies to \[ p \left (x \right ) = c_{1} x^{b} {\mathrm e}^{a x} \] Since \(p=\xi ^{\prime }\left (x \right )\) then the new first order ode to solve is \begin {align*} \xi ^{\prime }\left (x \right ) = c_{1} x^{b} {\mathrm e}^{a x} \end {align*}
Integrating both sides gives \begin {align*} \xi \left (x \right ) &= \int { c_{1} x^{b} {\mathrm e}^{a x}\,\mathop {\mathrm {d}x}}\\ &= -\frac {c_{1} \left (-a \right )^{-b} \left (x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )\right )}{a}+c_{2} \end {align*}
The original ode (2) now reduces to first order ode \begin {align*} \xi \left (x \right ) y^{\prime }-y \xi ^{\prime }\left (x \right )+\xi \left (x \right ) p \left (x \right ) y&=\int \xi \left (x \right ) r \left (x \right )d x\\ y^{\prime }+y \left (p \left (x \right )-\frac {\xi ^{\prime }\left (x \right )}{\xi \left (x \right )}\right )&=\frac {\int \xi \left (x \right ) r \left (x \right )d x}{\xi \left (x \right )}\\ y^{\prime }+y \left (\frac {a x +b}{x}+\frac {c_{1} \left (-a \right )^{-b} \left (-\frac {x^{b} b \left (-a \right )^{b} {\mathrm e}^{a x}}{x}-x^{b} \left (-a \right )^{b} a \,{\mathrm e}^{a x}-x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} a \left (-a x \right )^{-1+b} {\mathrm e}^{a x}\right )}{a \left (-\frac {c_{1} \left (-a \right )^{-b} \left (x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )\right )}{a}+c_{2} \right )}\right )&=0 \end {align*}
Which is now a first order ode. This is now solved for \(y\). In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {y \left (c_{1} \left (-a \right )^{-b} a \,x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \left (-a x \right )^{-1+b} {\mathrm e}^{a x} x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} a b x -\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} a b x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} b^{2}-\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} b^{2}-c_{2} a^{2} x -c_{2} a b \right )}{x \left (c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-c_{2} a \right )} \end {align*}
Where \(f(x)=-\frac {c_{1} \left (-a \right )^{-b} a \,x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \left (-a x \right )^{-1+b} {\mathrm e}^{a x} x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} a b x -\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} a b x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} b^{2}-\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} b^{2}-c_{2} a^{2} x -c_{2} a b}{x \left (c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-c_{2} a \right )}\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= -\frac {c_{1} \left (-a \right )^{-b} a \,x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \left (-a x \right )^{-1+b} {\mathrm e}^{a x} x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} a b x -\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} a b x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} b^{2}-\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} b^{2}-c_{2} a^{2} x -c_{2} a b}{x \left (c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-c_{2} a \right )} \,d x\\ \int { \frac {1}{y} \,dy} &= \int {-\frac {c_{1} \left (-a \right )^{-b} a \,x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \left (-a x \right )^{-1+b} {\mathrm e}^{a x} x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} a b x -\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} a b x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} b^{2}-\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} b^{2}-c_{2} a^{2} x -c_{2} a b}{x \left (c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-c_{2} a \right )} \,d x}\\ \ln \left (y \right )&=\int -\frac {c_{1} \left (-a \right )^{-b} a \,x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \left (-a x \right )^{-1+b} {\mathrm e}^{a x} x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} a b x -\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} a b x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} b^{2}-\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} b^{2}-c_{2} a^{2} x -c_{2} a b}{x \left (c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-c_{2} a \right )}d x +c_{3}\\ y&={\mathrm e}^{\int -\frac {c_{1} \left (-a \right )^{-b} a \,x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \left (-a x \right )^{-1+b} {\mathrm e}^{a x} x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} a b x -\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} a b x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} b^{2}-\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} b^{2}-c_{2} a^{2} x -c_{2} a b}{x \left (c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-c_{2} a \right )}d x +c_{3}}\\ &=c_{3} {\mathrm e}^{\int -\frac {c_{1} \left (-a \right )^{-b} a \,x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \left (-a x \right )^{-1+b} {\mathrm e}^{a x} x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} a b x -\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} a b x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} b^{2}-\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} b^{2}-c_{2} a^{2} x -c_{2} a b}{x \left (c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-c_{2} a \right )}d x} \end {align*}
Hence, the solution found using Lagrange adjoint equation method is \[ y = c_{3} {\mathrm e}^{\int -\frac {c_{1} \left (-a \right )^{-b} a \,x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \left (-a x \right )^{-1+b} {\mathrm e}^{a x} x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} a b x -\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} a b x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} b^{2}-\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} b^{2}-c_{2} a^{2} x -c_{2} a b}{x \left (c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-c_{2} a \right )}d x} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{3} {\mathrm e}^{\int -\frac {c_{1} \left (-a \right )^{-b} a \,x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \left (-a x \right )^{-1+b} {\mathrm e}^{a x} x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} a b x -\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} a b x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} b^{2}-\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} b^{2}-c_{2} a^{2} x -c_{2} a b}{x \left (c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-c_{2} a \right )}d x} \\ \end{align*}
Verification of solutions
\[ y = c_{3} {\mathrm e}^{\int -\frac {c_{1} \left (-a \right )^{-b} a \,x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \left (-a x \right )^{-1+b} {\mathrm e}^{a x} x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} a b x -\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} a b x +\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b \right ) c_{1} b^{2}-\left (-a \right )^{-b} x^{b} \left (-a \right )^{b} \left (-a x \right )^{-b} \Gamma \left (b , -a x \right ) c_{1} b^{2}-c_{2} a^{2} x -c_{2} a b}{x \left (c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \Gamma \left (b \right ) \left (-a x \right )^{-b}-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} b \left (-a x \right )^{-b} \Gamma \left (b , -a x \right )-c_{1} \left (-a \right )^{-b} x^{b} \left (-a \right )^{b} {\mathrm e}^{a x}-c_{2} a \right )}d x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d x}y^{\prime }\right )+\left (a x +b \right ) y^{\prime } x -b 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 {\left (a x +b \right ) y^{\prime }}{x}+\frac {b y}{x^{2}} \\ \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 +b \right ) y^{\prime }}{x}-\frac {b y}{x^{2}}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=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 +b}{x}, P_{3}\left (x \right )=-\frac {b}{x^{2}}\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}}}=b \\ {} & \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}}}=-b \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d x}y^{\prime }\right )+\left (a x +b \right ) y^{\prime } x -b 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^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..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^{2}\cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\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} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (r -1\right ) \left (b +r \right ) x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (a_{k} \left (k +r -1\right ) \left (k +b +r \right )+a a_{k -1} \left (k +r -1\right )\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & \left (r -1\right ) \left (b +r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{1, -b \right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (k +r -1\right ) \left (a_{k} \left (k +b +r \right )+a a_{k -1}\right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & \left (k +r \right ) \left (a_{k +1} \left (k +1+b +r \right )+a a_{k}\right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=-\frac {a a_{k}}{k +1+b +r} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =1 \\ {} & {} & a_{k +1}=-\frac {a a_{k}}{k +2+b} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =1 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +1}, a_{k +1}=-\frac {a a_{k}}{k +2+b}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-b \\ {} & {} & a_{k +1}=-\frac {a a_{k}}{k +1} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-b \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -b}, a_{k +1}=-\frac {a a_{k}}{k +1}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}c_{k} x^{k +1}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}d_{k} x^{k -b}\right ), c_{k +1}=-\frac {a c_{k}}{k +2+b}, d_{k +1}=-\frac {a d_{k}}{k +1}\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)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Reducible group (found an exponential solution) Group is reducible, not completely reducible <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 51
dsolve(x^2*diff(y(x),x$2)+(a*x^2+b*x)*diff(y(x),x)-b*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = -{\mathrm e}^{-a x} c_{2} \left (\Gamma \left (b , -a x \right ) b -\Gamma \left (b +1\right )\right ) \left (-a x \right )^{-b}+c_{1} x^{-b} {\mathrm e}^{-a x}-c_{2} \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 43
DSolve[x^2*y''[x]+(a*x^2+b*x)*y'[x]-b*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-a x} \left (\frac {c_1 (-a x)^{-b} \Gamma (b+1,-a x)}{a}+c_2 x^{-b}\right ) \]