Internal problem ID [10940]
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: 106.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a_{1} x +a_{0} \right ) y^{\prime \prime }+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 129
dsolve((a__1*x+a__0)*diff(y(x),x$2)+(b__1*x+b__0)*diff(y(x),x)-m*b__1*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {b_{1} x}{a_{1}}} \operatorname {KummerM}\left (m +1, \frac {a_{0} b_{1} +2 a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}, \frac {b_{1} \left (a_{1} x +a_{0} \right )}{a_{1}^{2}}\right ) \left (a_{1} x +a_{0} \right )^{\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}}+c_{2} {\mathrm e}^{-\frac {b_{1} x}{a_{1}}} \operatorname {KummerU}\left (m +1, \frac {a_{0} b_{1} +2 a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}, \frac {b_{1} \left (a_{1} x +a_{0} \right )}{a_{1}^{2}}\right ) \left (a_{1} x +a_{0} \right )^{\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}} \]
✓ Solution by Mathematica
Time used: 0.278 (sec). Leaf size: 102
DSolve[(a1*x+a0)*y''[x]+(b1*x+b0)*y'[x]-m*b1*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {\text {b1} x}{\text {a1}}} (\text {a0}+\text {a1} x)^{\frac {\text {a0} \text {b1}+\text {a1}^2-\text {a1} \text {b0}}{\text {a1}^2}} \left (c_1 \operatorname {HypergeometricU}\left (m+1,-\frac {\text {b0}}{\text {a1}}+\frac {\text {a0} \text {b1}}{\text {a1}^2}+2,\frac {\text {b1} (\text {a0}+\text {a1} x)}{\text {a1}^2}\right )+c_2 L_{-m-1}^{\frac {\text {a1}^2-\text {b0} \text {a1}+\text {a0} \text {b1}}{\text {a1}^2}}\left (\frac {\text {b1} (\text {a0}+\text {a1} x)}{\text {a1}^2}\right )\right ) \]