Suppose ( y_{1}(x)=e^{-2 x} ) and ( y_{2}(x)=e^{3 x} ) are solutions of some homogeneous second-order differential equation. (1) Show that ( y_{1}(x) ) and ( y_{2}(x) ) are linearly independent (you can do it in three ways :-). (2) Give the general solution of this equation. (3) Identify the solution satisfying initial conditions ( y(0)=3 ) and ( y^{prime}(0)=1 ).
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Given Y1 = -2x and Y2 = 3x, we need to find the Wronskian W(Y1, Y2) as follows: \[ W(Y1, Y2) = \begin{vmatrix} -2x & 3x \\ -2 & 3 \end{vmatrix} = (-2)(3) - (3x)(-2) = 6 + 6x = 6(1 + x) \] Show more…
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