76.12.23 problem 35

Internal problem ID [17579]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.2 (Theory of second order linear homogeneous equations). Problems at page 226
Problem number : 35
Date solved : Tuesday, January 28, 2025 at 10:44:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-\left (x -\frac {3}{16}\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x^{{1}/{4}} {\mathrm e}^{2 \sqrt {x}} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 25

dsolve([x^2*diff(y(x),x$2)-(x-1875/10000)*y(x)=0,x^(1/4)*exp(2*sqrt(x))],singsol=all)
 

\[ y = x^{{1}/{4}} \left (c_{1} \sinh \left (2 \sqrt {x}\right )+c_{2} \cosh \left (2 \sqrt {x}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.038 (sec). Leaf size: 41

DSolve[x^2*D[y[x],{x,2}]-(x-1875/10000)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} e^{-2 \sqrt {x}} \sqrt [4]{x} \left (2 c_1 e^{4 \sqrt {x}}-c_2\right ) \]