In this problem you will solve the nonhomogeneous differential y''+15y'+50y=sin(e^(5x)

This explains it step by step: https://www3.nd.edu/~apilking/Math10560/Lectures/22.%20Second%20order%20Nonhomogeneous%20Linear%20copy.pdf

I'd recommend using the Wronskian though to mitigate the amount of derivatives but you'll have to do integration by parts. The Wronskian usage is explained here: http://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx

I had a solution explained in immense detail but the site erased it because it took so long to compute and type the solution like all DE's.

for this case we have by definition, if two lines are perpendicular, then the product of their slopes is -1.

then, given the following line:

so,

we are looking for m_ {2}:

then, the line is given by:

we find "b" replacing the given point:

finally, the equation is:

option c