58.2.42 problem 42

Internal problem ID [9165]
Book : Second order enumerated odes
Section : section 2
Problem number : 42
Date solved : Monday, January 27, 2025 at 05:51:32 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-3 y^{\prime \prime }+5 y^{\prime }-2 y&=x \,{\mathrm e}^{x}+3 \,{\mathrm e}^{-2 x} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 52

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

\[ y = \frac {{\mathrm e}^{-2 x} \left (\left (x^{4}-\frac {4 x^{3}}{3}+\left (72 c_4 +\frac {4}{3}\right ) x^{2}+\left (72 c_3 -\frac {8}{9}\right ) x +72 c_{1} +\frac {8}{27}\right ) {\mathrm e}^{3 x}-8 x +72 c_{2} -8\right )}{72} \]

✓ Solution by Mathematica

Time used: 0.279 (sec). Leaf size: 170

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

\[ y(x)\to e^x x \int _1^x-\frac {1}{9} e^{-3 K[3]} (3 K[3]+1) \left (e^{3 K[3]} K[3]+3\right )dK[3]+e^{-2 x} \int _1^x\left (-\frac {1}{27} e^{3 K[1]} K[1]-\frac {1}{9}\right )dK[1]+e^x \int _1^x\frac {1}{54} e^{-3 K[2]} \left (e^{3 K[2]} K[2]+3\right ) \left (9 K[2]^2+6 K[2]+2\right )dK[2]+\frac {e^x x^4}{12}-\frac {1}{6} e^{-2 x} x^2+c_4 e^x x^2+c_3 e^x x+c_1 e^{-2 x}+c_2 e^x \]