82.27.4 problem Ex. 4

Internal problem ID [18879]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VI. Linear equations with constant coefficients. problems at page 73
Problem number : Ex. 4
Date solved : Tuesday, January 28, 2025 at 12:33:19 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }-8 y^{\prime }+12 y&=X \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 74

dsolve(diff(y(x),x$3)-diff(y(x),x$2)-8*diff(y(x),x)+12*y(x)=X(x),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\left (5 \left (\int X \left (x \right ) {\mathrm e}^{-2 x}d x \right ) {\mathrm e}^{5 x} x +25 c_3 \,{\mathrm e}^{5 x} x -\left (\int {\mathrm e}^{-2 x} \left (5 x +1\right ) X \left (x \right )d x \right ) {\mathrm e}^{5 x}+25 c_{2} {\mathrm e}^{5 x}+\int X \left (x \right ) {\mathrm e}^{3 x}d x +25 c_{1} \right ) {\mathrm e}^{-3 x}}{25} \]

✓ Solution by Mathematica

Time used: 0.045 (sec). Leaf size: 110

DSolve[D[y[x],{x,3}]-D[y[x],{x,2}]-8*D[y[x],x]+12*y[x]==X[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-3 x} \left (\int _1^x\frac {1}{25} e^{3 K[1]} X(K[1])dK[1]+e^{5 x} \int _1^x-\frac {1}{25} e^{-2 K[2]} (5 K[2]+1) X(K[2])dK[2]+e^{5 x} x \int _1^x\frac {1}{5} e^{-2 K[3]} X(K[3])dK[3]+c_2 e^{5 x}+c_3 e^{5 x} x+c_1\right ) \]