To solve the problem, let's go step-by-step through the provided information and the required calculations.
1. Particle's Equation of Motion:
Given the equation of motion of the particle:
[tex]\[
y = px - qx^2
\][/tex]
Here, [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are positive constants.
2. Velocity at the Origin:
Given the velocity of the particle at the origin:
[tex]\[
u = \sqrt{\frac{a(x + p^2)}{y q}}
\][/tex]
This expression gives us the velocity at the origin where [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex].
3. Formulate the Expression for [tex]\( x + y \)[/tex]:
To find [tex]\( x + y \)[/tex], we need an expression for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] from the given equation of motion.
The given equation of motion simplifies to:
[tex]\[
y = px - qx^2
\][/tex]
Substituting this into [tex]\( x + y \)[/tex] gives us:
[tex]\[
x + y = x + (px - qx^2)
\][/tex]
4. Combine the Terms:
Combine like terms in the expression:
[tex]\[
x + (px - qx^2) = x + px - qx^2
\][/tex]
5. Simplify the Expression:
Factor out the common term [tex]\( x \)[/tex]:
[tex]\[
x(1 + p - qx)
\][/tex]
6. Final Simplified Expression:
Hence, the expression for [tex]\( x + y \)[/tex]:
[tex]\[
x + y = x(p - qx) + x
\][/tex]
So the final answer is:
[tex]\[
x(p - qx) + x
\][/tex]
Thus, the value of [tex]\( x + y \)[/tex] can be expressed as [tex]\( x(p - qx) + x \)[/tex].
