83.17.8 problem 8

Internal problem ID [19204]
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 : 8
Date solved : Tuesday, January 28, 2025 at 01:13:23 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\left (6\right )}-2 y^{\left (5\right )}+3 y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+3 y^{\prime \prime }-2 y^{\prime }+y&=\sin \left (\frac {x}{2}\right )^{2}+{\mathrm e}^{x} \end{align*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 64

dsolve(diff(y(x),x$6)-2*diff(y(x),x$5)+3*diff(y(x),x$4)-4*diff(y(x),x$3)+3*diff(y(x),x$2)-2*diff(y(x),x)+y(x)=sin(x/2)^2+exp(x),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\left (4 x^{2}+\left (32 c_5 -16\right ) x +32 c_{2} +16\right ) {\mathrm e}^{x}}{32}+\frac {\left (-x^{2}+\left (32 c_6 -2\right ) x +32 c_3 +5\right ) \sin \left (x \right )}{32}+\frac {1}{2}+\frac {\left (\left (32 c_4 -5\right ) x +32 c_{1} -6\right ) \cos \left (x \right )}{32} \]

✓ Solution by Mathematica

Time used: 0.664 (sec). Leaf size: 76

DSolve[D[y[x],{x,6}]-2*D[y[x],{x,5}]+3*D[y[x],{x,4}]-4*D[y[x],{x,3}]+3*D[y[x],{x,2}]-2*D[y[x],x]+y[x]==Sin[x/2]^2+Exp[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{64} \left (8 \left (4+e^x \left (x^2-4 x+8 c_6 x+4+8 c_5\right )\right )+\left (-2 x^2+(-4+64 c_4) x+7+64 c_3\right ) \sin (x)+2 (4 (-1+8 c_2) x-5+32 c_1) \cos (x)\right ) \]