An Interesting Differential Equation

Calculus Level 3

Here is a differential equation:

x − 2 = x 3 e x d f d x x-2 = \frac{x^3}{e^x} \frac{df}{dx}

Given that f ( 1 ) = e f(1) = e , f ( 3 ) f(3) is of the form e a b \frac{e^a}{b} .

What is 2a+b?


The answer is 15.0.

Varun Gudibanda by Varun Gudibanda , 23, USA

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1 solution

Ahpa Tsum
Jan 4, 2019

by separate variables ,we get : e x . d x x 2 − 2. e x . d x x 3 = d f \frac{e^x.dx}{x^2} - \frac{2.e^x.dx}{x^3} = df by an easy observation we can see that the left side is the derivative of e x x 2 \frac{e^x}{x^2} hence, f ( x ) = e x x 2 + c f(x)= \frac{e^x}{x^2} + c by substituting , x=1 we have c=0 then, f ( 3 ) = e 3 9 f(3)=\frac{e^3}{9} . a=3 and b =9 finally we get 2a+b = 15

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