76.17.18 problem 27

Internal problem ID [17704]
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 4.7 (Variation of parameters). Problems at page 280
Problem number : 27
Date solved : Tuesday, January 28, 2025 at 11:01:31 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y&=g \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.228 (sec). Leaf size: 44

dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-1/4)*y(x)=g(x),y(x), singsol=all)
 

\[ y = \frac {\sin \left (x \right ) c_{2} +\cos \left (x \right ) c_{1} -\left (\int \frac {\sin \left (x \right ) g \left (x \right )}{x^{{3}/{2}}}d x \right ) \cos \left (x \right )+\left (\int \frac {\cos \left (x \right ) g \left (x \right )}{x^{{3}/{2}}}d x \right ) \sin \left (x \right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.076 (sec). Leaf size: 107

DSolve[x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-1/4)*y[x]==g[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^{-i x} \left (2 \int _1^x\frac {i e^{i K[1]} g(K[1])}{2 K[1]^{3/2}}dK[1]-i e^{2 i x} \int _1^x\frac {e^{-i K[2]} g(K[2])}{K[2]^{3/2}}dK[2]-i c_2 e^{2 i x}+2 c_1\right )}{2 \sqrt {x}} \]