Given differential equation is:(9x^3 8y/x)dx (y^2 8lnx)dy=0.
If a differential equation is of the form M(x,y)dx + N(x,y)dy = 0, then it is called an exact differential equation
if:∂M/∂y = ∂N/∂x
Here, M = 9x³ + 8y/x and N = y² + 8lnx.
Therefore, ∂M/∂y = 8 and ∂N/∂x = 8/x.
Thus, the given differential equation is an exact differential equation.
Now, to find the solution of an exact differential equation, we integrate either M or N with respect to x or y, respectively.
Let's integrate M w.r.t x. So, we get:
∫Mdx = ∫(9x³ + 8y/x)dx= 9/4 x⁴ + 8y ln x + h(y) (put h(y) = 0,
since ∂(∂M/∂y)/∂y = ∂(∂N/∂x)/∂x )
Differentiating the above w.r.t y, we get:(d/dy) ∫Mdx = 8x + h'(y)
Comparing the above with N = y² + 8lnx
We get, h'(y) = y²∴ h(y) = y³/3 + c Here, c is a constant of integration.
The general solution of is 9/4 x⁴ + 8y ln x + y³/3 = c.
Yes the differential equation is exact
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