35.7.23 problem 28

Internal problem ID [6205]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 7. Other second-Order equations. page 435
Problem number : 28
Date solved : Monday, January 27, 2025 at 01:47:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x y^{\prime \prime }-2 \left (3 x -1\right ) y^{\prime }+\left (3 x -2\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{x} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 14

dsolve([3*x*diff(y(x),x$2)-2*(3*x-1)*diff(y(x),x)+(3*x-2)*y(x)=0,exp(x)],singsol=all)
 

\[ y = {\mathrm e}^{x} \left (c_{1} +x^{{1}/{3}} c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 21

DSolve[3*x*D[y[x],{x,2}]-2*(3*x-1)*D[y[x],x]+(3*x-2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^x \left (3 c_2 \sqrt [3]{x}+c_1\right ) \]