Given: y" - 2y' = 6t + 5e^2t. Find the correct form to use for y_p if the equation is solved using Undetermined coefficients. Do NOT Solve the equation

Answer:[tex]y_p=A+Bt+Ce^{2t}[/tex]Step-by-step explanation:Given: [tex]y'' - 2y' = 6t + 5e^{2t}[/tex].we need to find the correct form for [tex]y_p[/tex] if the equation is solve using undetermined coefficients.A first order differential equation [tex]\frac{\mathrm{d} y}{\mathrm{d} x}=f\left ( x,y \right )[/tex] is said to be homogeneous if [tex]f(tx,ty)=f(x,y)[/tex] for all t.Consider homogeneous equation [tex]y''-2y'=0[/tex]Let [tex]y=e^{rt}[/tex] be the solution .We get [tex](r^2-2r)e^{rt}=0[/tex]Since [tex]e^{rt}\neq 0[/tex], [tex]r^2-2r=0[/tex].So, we get solution as [tex]y_c=c_1+c_2e^{2t}[/tex]As constant term and [tex]e^{2t}[/tex] are already in the R.H.S of equation [tex]y" - 2y' = 6t + 5e^{2t}[/tex], we can take [tex]y_p[/tex] as [tex]y_p=A+Bt+Ce^{2t}[/tex]