Internal problem ID [10973]
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: 139.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {a_{2} x^{2} y^{\prime \prime }+\left (a_{1} x^{2}+b_{1} x \right ) y^{\prime }+\left (x^{2} a_{0} +b_{0} x +c_{0} \right ) y=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 165
dsolve(a__2*x^2*diff(y(x),x$2)+(a__1*x^2+b__1*x)*diff(y(x),x)+(a__0*x^2+b__0*x+c__0)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {a_{1} x}{2 a_{2}}} x^{-\frac {b_{1}}{2 a_{2}}} \operatorname {WhittakerM}\left (-\frac {b_{1} a_{1} -2 a_{2} b_{0}}{2 a_{2} \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}}, \frac {\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, x}{a_{2}}\right )+c_{2} {\mathrm e}^{-\frac {a_{1} x}{2 a_{2}}} x^{-\frac {b_{1}}{2 a_{2}}} \operatorname {WhittakerW}\left (-\frac {b_{1} a_{1} -2 a_{2} b_{0}}{2 a_{2} \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}}, \frac {\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, x}{a_{2}}\right ) \]
✓ Solution by Mathematica
Time used: 0.538 (sec). Leaf size: 272
DSolve[a2*x^2*y''[x]+(a1*x^2+b1*x)*y'[x]+(a0*x^2+b0*x+c0)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {x \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}+\text {a1}\right )}{2 \text {a2}}} x^{\frac {\sqrt {\text {a2}^2-2 \text {a2} (\text {b1}+2 \text {c0})+\text {b1}^2}+\text {a2}-\text {b1}}{2 \text {a2}}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {-\frac {2 \text {b0} \text {a2}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\text {a2}+\frac {\text {a1} \text {b1}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{2 \text {a2}},\frac {\text {a2}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{\text {a2}},\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} x}{\text {a2}}\right )+c_2 L_{-\frac {-\frac {2 \text {b0} \text {a2}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\text {a2}+\frac {\text {a1} \text {b1}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{2 \text {a2}}}^{\frac {\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{\text {a2}}}\left (\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} x}{\text {a2}}\right )\right ) \]