12.9.33 problem 33

Internal problem ID [1789]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 33
Date solved : Monday, January 27, 2025 at 05:34:52 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \left (1+x \right )^{2} y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }-\left (x^{2}+2 x -1\right ) y&=\left (1+x \right )^{3} {\mathrm e}^{x} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\left (1+x \right ) {\mathrm e}^{x} \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 22

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

\[ y = \frac {\left (x +1\right ) \left (x \,{\mathrm e}^{x}+2 \cosh \left (x \right )-5 \sinh \left (x \right )\right )}{2} \]

✓ Solution by Mathematica

Time used: 21.293 (sec). Leaf size: 5749

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

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