Internal problem ID [10942]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-3 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 108.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a_{2} x +b_{2} \right ) y^{\prime \prime }+\left (a_{1} x +b_{1} \right ) y^{\prime }+\left (a_{0} x +b_{0} \right ) y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 287
dsolve((a__2*x+b__2)*diff(y(x),x$2)+(a__1*x+b__1)*diff(y(x),x)+(a__0*x+b__0)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {\left (\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}+a_{1} \right ) x}{2 a_{2}}} \operatorname {KummerM}\left (\frac {\left (a_{1} b_{2} +2 a_{2}^{2}-a_{2} b_{1} \right ) \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}-2 a_{2}^{2} b_{0} +\left (2 a_{0} b_{2} +b_{1} a_{1} \right ) a_{2} -a_{1}^{2} b_{2}}{2 \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, a_{2}^{2}}, \frac {a_{1} b_{2} +2 a_{2}^{2}-a_{2} b_{1}}{a_{2}^{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, \left (a_{2} x +b_{2} \right )}{a_{2}^{2}}\right ) \left (a_{2} x +b_{2} \right )^{\frac {a_{1} b_{2} +a_{2}^{2}-a_{2} b_{1}}{a_{2}^{2}}}+c_{2} {\mathrm e}^{-\frac {\left (\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}+a_{1} \right ) x}{2 a_{2}}} \operatorname {KummerU}\left (\frac {\left (a_{1} b_{2} +2 a_{2}^{2}-a_{2} b_{1} \right ) \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}-2 a_{2}^{2} b_{0} +\left (2 a_{0} b_{2} +b_{1} a_{1} \right ) a_{2} -a_{1}^{2} b_{2}}{2 \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, a_{2}^{2}}, \frac {a_{1} b_{2} +2 a_{2}^{2}-a_{2} b_{1}}{a_{2}^{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, \left (a_{2} x +b_{2} \right )}{a_{2}^{2}}\right ) \left (a_{2} x +b_{2} \right )^{\frac {a_{1} b_{2} +a_{2}^{2}-a_{2} b_{1}}{a_{2}^{2}}} \]
✓ Solution by Mathematica
Time used: 0.526 (sec). Leaf size: 301
DSolve[(a2*x+b2)*y''[x]+(a1*x+b1)*y'[x]+(a0*x+b0)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {x \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}+\text {a1}\right )}{2 \text {a2}}} (\text {a2} x+\text {b2})^{\frac {\text {a1} \text {b2}+\text {a2}^2-\text {a2} \text {b1}}{\text {a2}^2}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {2 \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}-\text {b0}\right ) \text {a2}^2+\left (\text {a1} \text {b1}-\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} \text {b1}+2 \text {a0} \text {b2}\right ) \text {a2}+\text {a1} \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}-\text {a1}\right ) \text {b2}}{2 \text {a2}^2 \sqrt {\text {a1}^2-4 \text {a0} \text {a2}}},-\frac {\text {b1}}{\text {a2}}+\frac {\text {a1} \text {b2}}{\text {a2}^2}+2,\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} (\text {b2}+\text {a2} x)}{\text {a2}^2}\right )+c_2 L_{\frac {-2 \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}-\text {b0}\right ) \text {a2}^2+\left (-\text {a1} \text {b1}+\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} \text {b1}-2 \text {a0} \text {b2}\right ) \text {a2}+\text {a1} \left (\text {a1}-\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}\right ) \text {b2}}{2 \text {a2}^2 \sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}}^{\frac {\text {a2}^2-\text {b1} \text {a2}+\text {a1} \text {b2}}{\text {a2}^2}}\left (\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} (\text {b2}+\text {a2} x)}{\text {a2}^2}\right )\right ) \]