# 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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