Question

Find the particular solution that satisfies the initial condition. (Enter your solution as an equation.) Differential Equation Initial Condition du dv

Answer 1

The answer is "
`u = e^{(9)/(2)}-(1)/(2) \cos v^2\\`"

Explanation:

In the given-question some of the equation is missing, which is defined as follows:

`\bold{(du)/(dv) = u v \sin v^2} \\\\ \bold{u(0) = e^4}`

Solution:

`\to (du)/(dv) = u v \sin v^2 \\\\ \to (du)/(u) = v \sin v^2 \ dv\\\\ \to \int (du)/(u) = \int v \sin v^2 \ dv +c\\\\`

`|n|u| = -(1)/(2) \cos v^2 +c\\\\\ given u(0) = e^4 \\\\\to |n (e^4) = -(1)/(2) \ cos (0) +c\\\\\to 4= -(1)/(2) +c\\\\\to c= 4+(1)/(2)\\\\\to c= (8+1)/(2)\\\\\to c= (9)/(2)\\\\`

`\to |n|u| =-(1)/(2) \cos v^2+(9)/(2)\\\\\\\boxed{u = e^{(9)/(2)}-(1)/(2) \cos v^2}\\`