# Wronskian of $e^{4 t}$, $e^{- \frac{7 t}{2}}$

The calculator will find the Wronskian of the $2$ functions $e^{4 t}$, $e^{- \frac{7 t}{2}}$, with steps shown.
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Calculate the Wronskian of $\left\{f_{1} = e^{4 t}, f_{2} = e^{- \frac{7 t}{2}}\right\}$.

### Solution

The Wronskian is given by the following determinant: $W{\left(f_{1},f_{2} \right)}\left(t\right) = \left|\begin{array}{cc}f_{1}\left(t\right) & f_{2}\left(t\right)\\f_{1}^{\prime}\left(t\right) & f_{2}^{\prime}\left(t\right)\end{array}\right|.$

In our case, $W{\left(f_{1},f_{2} \right)}\left(t\right) = \left|\begin{array}{cc}e^{4 t} & e^{- \frac{7 t}{2}}\\\left(e^{4 t}\right)^{\prime } & \left(e^{- \frac{7 t}{2}}\right)^{\prime }\end{array}\right|.$

Find the derivatives (for steps, see derivative calculator): $W{\left(f_{1},f_{2} \right)}\left(t\right) = \left|\begin{array}{cc}e^{4 t} & e^{- \frac{7 t}{2}}\\4 e^{4 t} & - \frac{7 e^{- \frac{7 t}{2}}}{2}\end{array}\right|.$

Find the determinant (for steps, see determinant calculator): $\left|\begin{array}{cc}e^{4 t} & e^{- \frac{7 t}{2}}\\4 e^{4 t} & - \frac{7 e^{- \frac{7 t}{2}}}{2}\end{array}\right| = - \frac{15 e^{\frac{t}{2}}}{2}.$

The Wronskian equals $- \frac{15 e^{\frac{t}{2}}}{2}$A.