Exponentially sinusoidal

Calculus Level 5

Consider the function

\[ f(y) = \displaystyle \int_{0} ^{\infty} [e^{(-x^2)} \sin(2xy)] \, dx \]

Let \(A\),\(B\),\(C\) and \(D\) respectively be the coefficients of \( \frac{d(f(y))}{dy}\), \(yf(y)\), \(y\) and \(y^0\) in the first order differential equation of \(f(y)\).

\(A\frac{d(f(y))}{dy}+ Byf(y)+Cy+D=0\) where it is also given that \(A\) is positive .

further \(f(y)\) can be expressed as follows

\[ f(y)= \int_{0} ^{y} [e^{(px^2)-(qy^2)}] \, dx \]

Then evaluate \(A+B+C+D+p+q \)

Note that \(A, B, C, D, p, q\) are real numbers and they have no common factor


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