75.16.8 problem 481

Internal problem ID [16981]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Trial and error method. Exercises page 132
Problem number : 481
Date solved : Tuesday, January 28, 2025 at 09:44:14 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }&=x \,{\mathrm e}^{4 x} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 25

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

\[ y = \frac {\left (8 x^{2}+16 c_{1} -4 x +1\right ) {\mathrm e}^{4 x}}{64}+c_{2} \]

✓ Solution by Mathematica

Time used: 1.989 (sec). Leaf size: 67

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

\begin{align*} y(x)\to \int _1^x\frac {1}{2} e^{4 K[1]} \left (K[1]^2+2 c_1\right )dK[1]+c_2 \\ y(x)\to \frac {1}{64} e^{4 x} \left (8 x^2-4 x+1\right )-\frac {5 e^4}{64}+c_2 \\ \end{align*}