62.31.1 problem Ex 1

Internal problem ID [12973]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 54. Change of independent variable. Page 127
Problem number : Ex 1
Date solved : Tuesday, January 28, 2025 at 04:45:56 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\left (2 \,{\mathrm e}^{x}-1\right ) y^{\prime }+{\mathrm e}^{2 x} y&={\mathrm e}^{4 x} \end{align*}

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 44

dsolve(diff(y(x),x$2)+(2*exp(x)-1)*diff(y(x),x)+exp(2*x)*y(x)=exp(4*x),y(x), singsol=all)
 

\[ y = {\mathrm e}^{\frac {x}{2}-{\mathrm e}^{x}} \sinh \left (\frac {x}{2}\right ) c_{2} +{\mathrm e}^{\frac {x}{2}-{\mathrm e}^{x}} \cosh \left (\frac {x}{2}\right ) c_{1} +{\mathrm e}^{2 x}-4 \,{\mathrm e}^{x}+6 \]

✓ Solution by Mathematica

Time used: 0.074 (sec). Leaf size: 64

DSolve[D[y[x],{x,2}]+(2*Exp[x]-1)*D[y[x],x]+Exp[2*x]*y[x]==Exp[4*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-e^x} \left (\int _1^x-e^{4 K[1]+e^{K[1]}}dK[1]+e^x \int _1^xe^{3 K[2]+e^{K[2]}}dK[2]+c_2 e^x+c_1\right ) \]