62.27.2 problem Ex 2

Internal problem ID [12940]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VII, Linear differential equations with constant coefficients. Article 50. Method of undetermined coefficients. Page 107
Problem number : Ex 2
Date solved : Tuesday, January 28, 2025 at 04:44:36 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+y&=2 x \,{\mathrm e}^{2 x}-\sin \left (x \right )^{2} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 35

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

\[ y = -\frac {1}{2}+2 \,{\mathrm e}^{2 x} \left (x -2\right )-\frac {3 \cos \left (2 x \right )}{50}-\frac {2 \sin \left (2 x \right )}{25}+\left (c_{1} x +c_{2} \right ) {\mathrm e}^{x} \]

✓ Solution by Mathematica

Time used: 0.524 (sec). Leaf size: 83

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

\[ y(x)\to e^x \left (x \int _1^x\left (2 e^{K[2]} K[2]-e^{-K[2]} \sin ^2(K[2])\right )dK[2]+\int _1^x-\frac {1}{2} e^{-K[1]} K[1] \left (\cos (2 K[1])+4 e^{2 K[1]} K[1]-1\right )dK[1]+c_2 x+c_1\right ) \]