User:Kongrui875/report4

Problem 2: Solve an N2-ODE
Report problem 4.2 from.

Find: solution for the N2-ODE
1. Find $$m,n \in \mathbb{R}$$ that ($$)is exact.

2. Show that the first integral of ($$) is a L1-ODE-VC

where

3. Solve for y(x)

Solution
1. The first exactness condition of N2-ODE is

Rearrange ($$) to get

Thus, it satisfies the first exactness condition, where

The second exactness condition is

where

thus, from ($$),

from ($$),

if ($$) is true for any x and y,

substitute ($$) into ($$),

thus, Thus, $$m=\frac{1}{2}, n=0$$ for ($$) to be exact.

2.

integral both sides of ($$)

then

compare with ($$),

Assume

so

thus, the first integral of ($$) is a L1-ODE-VC shown in ($$)

3. Rearrange ($$) to get

it is in the form of