64.11.26 problem 26

Internal problem ID [13476]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number : 26
Date solved : Tuesday, January 28, 2025 at 05:44:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+4 y&=16 x +20 \,{\mathrm e}^{x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=3 \end{align*}

✓ Solution by Maple

Time used: 0.022 (sec). Leaf size: 19

dsolve([diff(y(x),x$2)+5*diff(y(x),x)+4*y(x)=16*x+20*exp(x),y(0) = 0, D(y)(0) = 3],y(x), singsol=all)
 

\[ y = 3 \,{\mathrm e}^{-x}-5+2 \,{\mathrm e}^{x}+4 x \]

✓ Solution by Mathematica

Time used: 0.348 (sec). Leaf size: 141

DSolve[{D[y[x],{x,2}]+5*D[y[x],x]+4*y[x]==16*x+20*Exp[x],{y[0]==0,Derivative[1][y][0] ==3}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-4 x} \left (\int _1^x-\frac {4}{3} e^{4 K[1]} \left (4 K[1]+5 e^{K[1]}\right )dK[1]-e^{3 x} \int _1^0\frac {4}{3} e^{K[2]} \left (4 K[2]+5 e^{K[2]}\right )dK[2]+e^{3 x} \int _1^x\frac {4}{3} e^{K[2]} \left (4 K[2]+5 e^{K[2]}\right )dK[2]-\int _1^0-\frac {4}{3} e^{4 K[1]} \left (4 K[1]+5 e^{K[1]}\right )dK[1]+e^{3 x}-1\right ) \]