Internal problem ID [8876]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1296.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\mathit {a2} \,x^{2} y^{\prime \prime }+\left (x^{2} \mathit {a1} +x \mathit {b1} \right ) y^{\prime }+\left (\mathit {a0} \,x^{2}+\mathit {b0} x +\mathit {c0} \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 165
dsolve(a2*x^2*diff(diff(y(x),x),x)+(a1*x^2+b1*x)*diff(y(x),x)+(a0*x^2+b0*x+c0)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{-\frac {\mathit {b1}}{2 \mathit {a2}}} {\mathrm e}^{-\frac {\mathit {a1} x}{2 \mathit {a2}}} \WhittakerM \left (-\frac {\mathit {a1} \mathit {b1} -2 \mathit {a2} \mathit {b0}}{2 \mathit {a2} \sqrt {-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}}, \frac {\sqrt {\mathit {a2}^{2}+\left (-2 \mathit {b1} -4 \mathit {c0} \right ) \mathit {a2} +\mathit {b1}^{2}}}{2 \mathit {a2}}, \frac {\sqrt {-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}\, x}{\mathit {a2}}\right )+c_{2} x^{-\frac {\mathit {b1}}{2 \mathit {a2}}} {\mathrm e}^{-\frac {\mathit {a1} x}{2 \mathit {a2}}} \WhittakerW \left (-\frac {\mathit {a1} \mathit {b1} -2 \mathit {a2} \mathit {b0}}{2 \mathit {a2} \sqrt {-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}}, \frac {\sqrt {\mathit {a2}^{2}+\left (-2 \mathit {b1} -4 \mathit {c0} \right ) \mathit {a2} +\mathit {b1}^{2}}}{2 \mathit {a2}}, \frac {\sqrt {-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}\, x}{\mathit {a2}}\right ) \]
✓ Solution by Mathematica
Time used: 0.213 (sec). Leaf size: 257
DSolve[(c0 + b0*x + a0*x^2)*y[x] + (b1*x + a1*x^2)*y'[x] + a2*x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\frac {x \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}+\text {a1}\right )}{2 \text {a2}}} x^{\frac {\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}+\text {a2}-\text {b1}}{2 \text {a2}}} \left (c_1 \text {HypergeometricU}\left (\frac {-\frac {2 \text {a2} \text {b0}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\frac {\text {a1} \text {b1}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}+\text {a2}}{2 \text {a2}},\frac {\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}+\text {a2}}{\text {a2}},\frac {x \sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}{\text {a2}}\right )+c_2 L_{-\frac {-\frac {2 \text {b0} \text {a2}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\text {a2}+\frac {\text {a1} \text {b1}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}}{2 \text {a2}}}^{\frac {\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}}{\text {a2}}}\left (\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} x}{\text {a2}}\right )\right ) \\ \end{align*}