75.16.45 problem 518

Internal problem ID [17018]
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 : 518
Date solved : Tuesday, January 28, 2025 at 09:45:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+4 y&=x^{2} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 24

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

\[ y = \frac {3}{8}+\left (c_{1} x +c_{2} \right ) {\mathrm e}^{2 x}+\frac {x^{2}}{4}+\frac {x}{2} \]

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 37

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

\[ y(x)\to \frac {1}{8} \left (2 x^2+4 x+3\right )+c_1 e^{2 x}+c_2 e^{2 x} x \]