Equation of the Streamline Passing Through Point (4,3)

To find the equation of the streamline passing through the point (4,3), we need to consider the given velocity equation and apply the concept of streamlines.

Understanding the Given Velocity Equation
The velocity equation is given as V = 2xi + 3yj, where i and j are the unit vectors along the x and y directions, respectively. This equation represents the velocity of the fluid flow at any point in the x-y plane.

The velocity vector V can be separated into its x and y components as follows:
Vx = 2x
Vy = 3y

Concept of Streamlines
Streamlines are imaginary lines that represent the path followed by fluid particles in a flow. These lines are always tangent to the velocity vector at any given point. In other words, the velocity vector at each point on a streamline is parallel to the streamline itself.

Steps to Find the Equation of the Streamline
1. Start by considering a point (x, y) on the streamline passing through (4,3).
2. At this point, the velocity vector V will be tangent to the streamline.
3. The velocity vector V can be written as V = Vx i + Vy j.
4. Since the velocity vector is tangent to the streamline, it must be parallel to the tangent vector of the streamline at that point.
5. The tangent vector to the streamline passing through (x, y) can be represented as T = dx i + dy j, where dx and dy are small increments in the x and y directions.
6. Since V is parallel to T, we can equate the components of V and T to find the relationship between dx and dy.
Vx = dx
Vy = dy
7. From step 6, we have:
dx = 2x
dy = 3y
8. Solve the differential equations from step 7 to find the relationship between x and y. This will give us the equation of the streamline passing through (4,3).

Solving the Differential Equations
Let's solve the differential equations from step 7 to find the equation of the streamline:
dx/dy = (2x)/(3y)

Separating variables and integrating:
∫(1/x)dx = ∫(2/3y)dy

ln|x| = (2/3)ln|y| + C

Using logarithmic properties, we can simplify the equation:
ln|x| - (2/3)ln|y| = C

Exponentiating both sides:
|x|/(|y|^2/3) = e^C

Simplifying further and substituting the point (4,3):
|x|/(|y|^2/3) = k, where k is a constant

Since |x| and |y| are always positive, we can remove the absolute value signs:
x/(y^(2/3)) = k

Therefore, the equation of the streamline passing through the point (4,3) is:
x/(y^(2/3)) = k

Conclusion
The equation of the streamline passing through the point (4,3) is x/(y^(2/3)) = k. This equation represents the path followed by fluid particles