Final answer:
To solve the given differential equation (x + 2y)dv+ (2x+ y)dy = 0, we can convert it to separable form. This involves rearranging and separating the variables. The solution for v can be obtained by integrating and simplifying the equation, followed by taking the derivative of the expression with respect to x.
Step-by-step explanation:
To solve the given differential equation (x + 2y)dv+ (2x+ y)dy = 0, we can convert it to separable form. Rearranging the equation, we get:
(x + 2y)dv = - (2x+ y)dy
Next, we divide both sides of the equation by (2x+ y)(x + 2y). This allows us to separate the variables:
dv/(2x+ y) = - dy/(x + 2y)
Now, we can integrate both sides of the equation:
∫ dv/(2x+ y) = - ∫ dy/(x + 2y)
This gives us:
ln|2x+ y| = -ln|x + 2y| + C
where C is the constant of integration.
To solve for v, we can eliminate the natural logarithms by taking the exponential of both sides:
|2x+ y| = 1/(|x + 2y|) * e^C
Simplifying further, we get:
2x+ y = ± e^C / |x + 2y|
Finally, we can solve for v by taking the derivative of the expression with respect to x:
v = d(± e^C / |x + 2y|) / dx