83.36.10 problem 10

Internal problem ID [19444]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise VIII (A) at page 125
Problem number : 10
Date solved : Tuesday, January 28, 2025 at 01:43:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 27

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

\[ y \left (x \right ) = {\mathrm e}^{a x} \left (c_{1} +\operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {-a}\, \left (x -2\right )}{2}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.171 (sec). Leaf size: 54

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

\[ y(x)\to \frac {1}{2} e^{a x} \left (\frac {\sqrt {2 \pi } e^{-2 a} c_2 \text {erfi}\left (\frac {\sqrt {a} (x-2)}{\sqrt {2}}\right )}{\sqrt {a}}+2 c_1\right ) \]