Wronskian of $$$e^{- 3 x}$$$, $$$e^{- x}$$$, $$$e^{6 x}$$$
Your Input
Calculate the Wronskian of $$$\left\{f_{1} = e^{- 3 x}, f_{2} = e^{- x}, f_{3} = e^{6 x}\right\}$$$.
Solution
The Wronskian is given by the following determinant: $$$W{\left(f_{1},f_{2},f_{3} \right)}\left(x\right) = \left|\begin{array}{ccc}f_{1}\left(x\right) & f_{2}\left(x\right) & f_{3}\left(x\right)\\f_{1}^{\prime}\left(x\right) & f_{2}^{\prime}\left(x\right) & f_{3}^{\prime}\left(x\right)\\f_{1}^{\prime\prime}\left(x\right) & f_{2}^{\prime\prime}\left(x\right) & f_{3}^{\prime\prime}\left(x\right)\end{array}\right|.$$$
In our case, $$$W{\left(f_{1},f_{2},f_{3} \right)}\left(x\right) = \left|\begin{array}{ccc}e^{- 3 x} & e^{- x} & e^{6 x}\\\left(e^{- 3 x}\right)^{\prime } & \left(e^{- x}\right)^{\prime } & \left(e^{6 x}\right)^{\prime }\\\left(e^{- 3 x}\right)^{\prime \prime } & \left(e^{- x}\right)^{\prime \prime } & \left(e^{6 x}\right)^{\prime \prime }\end{array}\right|.$$$
Find the derivatives (for steps, see derivative calculator): $$$W{\left(f_{1},f_{2},f_{3} \right)}\left(x\right) = \left|\begin{array}{ccc}e^{- 3 x} & e^{- x} & e^{6 x}\\- 3 e^{- 3 x} & - e^{- x} & 6 e^{6 x}\\9 e^{- 3 x} & e^{- x} & 36 e^{6 x}\end{array}\right|.$$$
Find the determinant (for steps, see determinant calculator): $$$\left|\begin{array}{ccc}e^{- 3 x} & e^{- x} & e^{6 x}\\- 3 e^{- 3 x} & - e^{- x} & 6 e^{6 x}\\9 e^{- 3 x} & e^{- x} & 36 e^{6 x}\end{array}\right| = 126 e^{2 x}.$$$
Answer
The Wronskian equals $$$126 e^{2 x}$$$A.