83.16.6 problem 6

Internal problem ID [19195]
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. Exercise III (H) at page 47
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 01:13:01 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+y^{\prime \prime }+y&=a \,x^{2}+b \,{\mathrm e}^{-x} \sin \left (2 x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.820 (sec). Leaf size: 75

dsolve(diff(y(x),x$4)+diff(y(x),x$2)+y(x)=a*x^2+b*exp(-x)*sin(2*x),y(x), singsol=all)
 

\[ y \left (x \right ) = \left (\left ({\mathrm e}^{\frac {x}{2}} c_{1} +c_3 \,{\mathrm e}^{\frac {3 x}{2}}\right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\left ({\mathrm e}^{\frac {x}{2}} c_{2} +c_4 \,{\mathrm e}^{\frac {3 x}{2}}\right ) \sin \left (\frac {\sqrt {3}\, x}{2}\right )-\frac {20 \cos \left (2 x \right ) b}{481}-\frac {9 b \sin \left (2 x \right )}{481}+a \,{\mathrm e}^{x} \left (x^{2}-2\right )\right ) {\mathrm e}^{-x} \]

✓ Solution by Mathematica

Time used: 3.186 (sec). Leaf size: 112

DSolve[D[y[x],{x,4}]+D[y[x],{x,2}]+y[x]==a*x^2+b*Exp[-x]*Sin[2*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to a x^2-2 a-\frac {9}{481} b e^{-x} \sin (2 x)-\frac {20}{481} b e^{-x} \cos (2 x)+e^{-x/2} \left (c_2 e^x+c_4\right ) \cos \left (\frac {\sqrt {3} x}{2}\right )+c_1 e^{-x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )+c_3 e^{x/2} \sin \left (\frac {\sqrt {3} x}{2}\right ) \]