64.11.36 problem 36

Internal problem ID [13486]
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 : 36
Date solved : Tuesday, January 28, 2025 at 05:45:35 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 30

dsolve([diff(y(x),x$2)-y(x)=3*x^2*exp(x),y(0) = 1, D(y)(0) = 2],y(x), singsol=all)
 

\[ y = -\frac {{\mathrm e}^{-x}}{8}+\frac {\left (4 x^{3}-6 x^{2}+6 x +9\right ) {\mathrm e}^{x}}{8} \]

✓ Solution by Mathematica

Time used: 0.045 (sec). Leaf size: 78

DSolve[{D[y[x],{x,2}]-y[x]==3*x^2*Exp[x],{y[0]==1,Derivative[1][y][0] ==2}},y[x],x,IncludeSingularSolutions -> True]
 

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