Internal problem ID [8725]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1145.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (\mathit {a2} x +\mathit {b2} \right ) y^{\prime \prime }+\left (\mathit {a1} x +\mathit {b1} \right ) y^{\prime }+\left (\mathit {a0} x +\mathit {b0} \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 287
dsolve((a2*x+b2)*diff(diff(y(x),x),x)+(a1*x+b1)*diff(y(x),x)+(a0*x+b0)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {\left (\sqrt {-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}+\mathit {a1} \right ) x}{2 \mathit {a2}}} \left (\mathit {a2} x +\mathit {b2} \right )^{\frac {\mathit {a1} \mathit {b2} +\mathit {a2}^{2}-\mathit {a2} \mathit {b1}}{\mathit {a2}^{2}}} \KummerM \left (\frac {\left (\mathit {a1} \mathit {b2} +2 \mathit {a2}^{2}-\mathit {a2} \mathit {b1} \right ) \sqrt {-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}-2 \mathit {a2}^{2} \mathit {b0} +\left (2 \mathit {a0} \mathit {b2} +\mathit {a1} \mathit {b1} \right ) \mathit {a2} -\mathit {a1}^{2} \mathit {b2}}{2 \sqrt {-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}\, \mathit {a2}^{2}}, \frac {\mathit {a1} \mathit {b2} +2 \mathit {a2}^{2}-\mathit {a2} \mathit {b1}}{\mathit {a2}^{2}}, \frac {\sqrt {-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}\, \left (\mathit {a2} x +\mathit {b2} \right )}{\mathit {a2}^{2}}\right )+c_{2} {\mathrm e}^{-\frac {\left (\sqrt {-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}+\mathit {a1} \right ) x}{2 \mathit {a2}}} \left (\mathit {a2} x +\mathit {b2} \right )^{\frac {\mathit {a1} \mathit {b2} +\mathit {a2}^{2}-\mathit {a2} \mathit {b1}}{\mathit {a2}^{2}}} \KummerU \left (\frac {\left (\mathit {a1} \mathit {b2} +2 \mathit {a2}^{2}-\mathit {a2} \mathit {b1} \right ) \sqrt {-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}-2 \mathit {a2}^{2} \mathit {b0} +\left (2 \mathit {a0} \mathit {b2} +\mathit {a1} \mathit {b1} \right ) \mathit {a2} -\mathit {a1}^{2} \mathit {b2}}{2 \sqrt {-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}\, \mathit {a2}^{2}}, \frac {\mathit {a1} \mathit {b2} +2 \mathit {a2}^{2}-\mathit {a2} \mathit {b1}}{\mathit {a2}^{2}}, \frac {\sqrt {-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}}\, \left (\mathit {a2} x +\mathit {b2} \right )}{\mathit {a2}^{2}}\right ) \]
✓ Solution by Mathematica
Time used: 0.179 (sec). Leaf size: 301
DSolve[(b0 + a0*x)*y[x] + (b1 + a1*x)*y'[x] + (b2 + a2*x)*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}}} (\text {a2} x+\text {b2})^{\frac {\text {a1} \text {b2}+\text {a2}^2-\text {a2} \text {b1}}{\text {a2}^2}} \left (c_1 \text {HypergeometricU}\left (\frac {2 \text {a2}^2 \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}-\text {b0}\right )+\text {a2} \left (-\text {b1} \sqrt {\text {a1}^2-4 \text {a0} \text {a2}}+2 \text {a0} \text {b2}+\text {a1} \text {b1}\right )+\text {a1} \text {b2} \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}-\text {a1}\right )}{2 \text {a2}^2 \sqrt {\text {a1}^2-4 \text {a0} \text {a2}}},\frac {\text {a1} \text {b2}-\text {a2} \text {b1}}{\text {a2}^2}+2,\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} (\text {a2} x+\text {b2})}{\text {a2}^2}\right )+c_2 L_{\frac {2 \left (\text {b0}-\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}\right ) \text {a2}^2+\left (-\text {a1} \text {b1}+\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} \text {b1}-2 \text {a0} \text {b2}\right ) \text {a2}+\text {a1} \left (\text {a1}-\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}\right ) \text {b2}}{2 \text {a2}^2 \sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}}^{\frac {\text {a2}^2-\text {b1} \text {a2}+\text {a1} \text {b2}}{\text {a2}^2}}\left (\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} (\text {b2}+\text {a2} x)}{\text {a2}^2}\right )\right ) \\ \end{align*}