82.39.13 problem Ex. 13

Internal problem ID [18945]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VII. Linear equations with variable coefficients. End of chapter problems at page 91
Problem number : Ex. 13
Date solved : Tuesday, January 28, 2025 at 12:37:39 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 35

dsolve(x^2*diff(y(x),x$2)-(2*m-1)*x*diff(y(x),x)+(m^2+n^2)*y(x)=n^2*x^m*ln(x),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {x^{m} \left (\left (c_{2} n -1\right ) \sin \left (n \ln \left (x \right )\right )+n \left (\cos \left (n \ln \left (x \right )\right ) c_{1} +\ln \left (x \right )\right )\right )}{n} \]

✓ Solution by Mathematica

Time used: 0.367 (sec). Leaf size: 79

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

\[ y(x)\to x^m \left (c_1 x^{-\frac {n \sqrt {m^2+n^2}}{\sqrt {-m^2-n^2}}}+c_2 x^{\frac {n \sqrt {m^2+n^2}}{\sqrt {-m^2-n^2}}}+\log (x)\right ) \]