61.28.46 problem 106

Internal problem ID [12606]
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
Problem number : 106
Date solved : Tuesday, January 28, 2025 at 08:02:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a_{1} x +a_{0} \right ) y^{\prime \prime }+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.184 (sec). Leaf size: 101

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)
 

\[ y = {\mathrm e}^{-\frac {b_{1} x}{a_{1}}} \left (a_{1} x +a_{0} \right )^{\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}} \left (\operatorname {KummerM}\left (m +1, \frac {a_{0} b_{1} +2 a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}, \frac {b_{1} \left (a_{1} x +a_{0} \right )}{a_{1}^{2}}\right ) c_{1} +\operatorname {KummerU}\left (m +1, \frac {a_{0} b_{1} +2 a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}, \frac {b_{1} \left (a_{1} x +a_{0} \right )}{a_{1}^{2}}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 1.124 (sec). Leaf size: 100

DSolve[(a1*x+a0)*D[y[x],{x,2}]+(b1*x+b0)*D[y[x],x]-m*b1*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (c_1 \operatorname {HypergeometricU}\left (m+1,-\frac {\text {b0}}{\text {a1}}+\frac {\text {a0} \text {b1}}{\text {a1}^2}+2,\frac {\text {b1} (\text {a0}+\text {a1} x)}{\text {a1}^2}\right )+c_2 L_{-m-1}^{\frac {\text {a1}^2-\text {b0} \text {a1}+\text {a0} \text {b1}}{\text {a1}^2}}\left (\frac {\text {b1} (\text {a0}+\text {a1} x)}{\text {a1}^2}\right )\right ) \exp \left (\int _1^x\frac {\text {a1}-\text {b0}-\text {b1} K[1]}{\text {a0}+\text {a1} K[1]}dK[1]\right ) \]