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Find the particular solution for
when
Solution:
If you examine the given equation, it is a differential equation because it has dy and dx in the equation. The type of equation is Homogeneous because the functions and variables cannot be separated by Separation of Variables. There's a method to solve the Homogeneous Functions. Consider the given equation
Let y = vx
dy = vdx + xdv
Substitute y and dy to the above equation, we have
The above equation can now be separated by Separation of Variables. Arrange the equation according to their variables
Integrate both sides of the equation
Take the inverse natural logarithm on both sides of the equation
but y = vx and v = y/x,
To solve for C, substitute the following:
to the above equation, we have
Therefore,
