x′−y−y′=−etdxdt−y−dydt=−etIfy=x=emt,dydt=dxdt=memt ⟹ memt−emt−memt=−et−emt=−et ⟹ m=1m=1is the only solutionto the equation.∴y=x=etis a solution to the firstorder linear differential equation.x' - y - y' = -e^t\\ \displaystyle\frac{\mathrm{d}x}{\mathrm{d}t} - y - \frac{\mathrm{d}y}{\mathrm{d}t} = -e^t\\ \textsf{If}\hspace{0.1cm} y = x = e^{mt}, \frac{\mathrm{d}y}{\mathrm{d}t} = \frac{\mathrm{d}x}{\mathrm{d}t} = me^{mt}\\ \implies me^{mt} - e^{mt} - me^{mt}= -e^t\\ -e^{mt} = -e^{t} \implies m = 1\\ m = 1\hspace{0.1cm}\textsf{is the only solution}\\\textsf{to the equation.}\\ \therefore y = x = e^{t} \hspace{0.1cm}\textsf{is a solution to the first}\\\textsf{order linear differential equation.}x′−y−y′=−etdtdx−y−dtdy=−etIfy=x=emt,dtdy=dtdx=memt⟹memt−emt−memt=−et−emt=−et⟹m=1m=1is the only solutionto the equation.∴y=x=etis a solution to the firstorder linear differential equation.
Need a fast expert's response?
and get a quick answer at the best price
for any assignment or question with DETAILED EXPLANATIONS!
Comments