83.17.9 problem 9

Internal problem ID [19205]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter III. Ordinary linear differential equations with constant coefficients. Misc. Examples on chapter III at page 50
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 01:13:25 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime \prime }+y^{\prime \prime }+16 y&=16 x^{2}+256 \end{align*}

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 57

dsolve(diff(y(x),x$4)+diff(y(x),x$2)+16*y(x)=16*x^2+256,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\left (8 \cos \left (\frac {3 x}{2}\right ) c_{1} +8 \sin \left (\frac {3 x}{2}\right ) c_{2} \right ) {\mathrm e}^{-\frac {\sqrt {7}\, x}{2}}}{8}+\frac {\left (8 \cos \left (\frac {3 x}{2}\right ) c_3 +8 \sin \left (\frac {3 x}{2}\right ) c_4 \right ) {\mathrm e}^{\frac {\sqrt {7}\, x}{2}}}{8}+x^{2}+\frac {127}{8} \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 132

DSolve[D[y[x],{x,4}]+D[y[x],{x,2}]+16*y[x]==16*x^2+256,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-2 x \sin \left (\frac {1}{2} \arctan \left (3 \sqrt {7}\right )\right )} \cos \left (2 x \cos \left (\frac {1}{2} \arctan \left (3 \sqrt {7}\right )\right )\right ) \left (c_3 e^{4 x \sin \left (\frac {1}{2} \arctan \left (3 \sqrt {7}\right )\right )}+c_2\right )+e^{-2 x \sin \left (\frac {1}{2} \arctan \left (3 \sqrt {7}\right )\right )} \left (c_1 e^{4 x \sin \left (\frac {1}{2} \arctan \left (3 \sqrt {7}\right )\right )}+c_4\right ) \sin \left (2 x \cos \left (\frac {1}{2} \arctan \left (3 \sqrt {7}\right )\right )\right )+x^2+\frac {127}{8} \]