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Wronskiano de $$$e^{4 t}$$$, $$$e^{- \frac{7 t}{2}}$$$
Sua entrada
Calcule o Wronskiano de $$$\left\{f_{1} = e^{4 t}, f_{2} = e^{- \frac{7 t}{2}}\right\}$$$.
Solução
O Wronskiano é dado pelo seguinte determinante: $$$W{\left(f_{1},f_{2} \right)}\left(t\right) = \left|\begin{array}{cc}f_{1}\left(t\right) & f_{2}\left(t\right)\\f_{1}^{\prime}\left(t\right) & f_{2}^{\prime}\left(t\right)\end{array}\right|.$$$
No nosso caso, $$$W{\left(f_{1},f_{2} \right)}\left(t\right) = \left|\begin{array}{cc}e^{4 t} & e^{- \frac{7 t}{2}}\\\left(e^{4 t}\right)^{\prime } & \left(e^{- \frac{7 t}{2}}\right)^{\prime }\end{array}\right|.$$$
Encontre as derivadas (para os passos, veja calculadora de derivadas): $$$W{\left(f_{1},f_{2} \right)}\left(t\right) = \left|\begin{array}{cc}e^{4 t} & e^{- \frac{7 t}{2}}\\4 e^{4 t} & - \frac{7 e^{- \frac{7 t}{2}}}{2}\end{array}\right|$$$
Calcule o determinante (para as etapas, consulte calculadora de determinante): $$$\left|\begin{array}{cc}e^{4 t} & e^{- \frac{7 t}{2}}\\4 e^{4 t} & - \frac{7 e^{- \frac{7 t}{2}}}{2}\end{array}\right| = - \frac{15 e^{\frac{t}{2}}}{2}.$$$
Resposta
O wronskiano é igual a $$$- \frac{15 e^{\frac{t}{2}}}{2}$$$A.