Show that every member of the family of function y = frac1 + cet1 - ce | Mathematics I 2081

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Mathematics I20815 marksNumerical: differential equation verificationIntroduction to first order equations Separable equations

Show that every member of the family of function $y = \frac{1 + ce^t}{1 - ce^t}$ is a solution of the differential equation $y' = \frac{1}{2}(y^2 - 1)$ . [5]

Show that every member of the family of function $y = \frac{1 + ce^t}{1 - ce^t}$ is a solution of the differential equation $y' = \frac{1}{2}(y^2 - 1)$. [5]

Family of functions: $y = \dfrac{1 + ce^t}{1 - ce^t}$ - Differential equation: $y' = \dfrac{1}{2}(y^2 - 1)$ Using the quotient rule with $u = 1 + ce^t$ , $v = 1 - ce^t$ : $u' = ce^t, \qquad v' = -ce^t$ $y' = \frac{u'v - uv'}{v^2} = \frac{ce^t(1 - ce^t) - (1 + ce^t)(-ce^t)}{(1 - ce^t)^2}$ Simp...

Family of functions: $y = \dfrac{1 + ce^t}{1 - ce^t}$ - Differential equation: $y' = \dfrac{1}{2}(y^2 - 1)$ Using the quotient rule with $u = 1 + ce^t$, $v = 1 - ce^t$: $$u' = ce^t, \qquad v' = -ce^t$$ $$y' = \frac{u'v - uv'}{v^2} = \frac{ce^t(1 - ce^t) - (1 + ce^t)(-ce^t)}{(1 - ce^t)^2}$$ Simp...

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Show that every member of the family of function y = frac1 + cet1 - cet is a solution of the differential equation y' = frac12(y2 - 1).