31.1 problem Ex 1

Internal problem ID [10283]

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.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]

Solve \begin {gather*} \boxed {y^{\prime \prime }+\left (2 \,{\mathrm e}^{x}-1\right ) y^{\prime }+{\mathrm e}^{2 x} y-{\mathrm e}^{4 x}=0} \end {gather*}

Solution by Maple

Time used: 0.015 (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)
 

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

Solution by Mathematica

Time used: 0.03 (sec). Leaf size: 36

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

\begin{align*} y(x)\to 6+e^{-e^x} \left (e^x \left (e^{e^x} \left (e^x-4\right )+c_2\right )+c_1\right ) \\ \end{align*}