35.7.22 problem 27

Internal problem ID [6204]
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 : 27
Date solved : Monday, January 27, 2025 at 01:47:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+\left (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: 0.003 (sec). Leaf size: 14

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

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

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 25

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

\[ y(x)\to \frac {1}{3} e^{x+1} \left (c_2 x^3+3 c_1\right ) \]