User:Egm6321.f10.team6.lee/HW4

= Problem 1. Solve N2_ODE_VC =

Required
1. show eqn(1.1) is exact

2. find $$\begin{align}\phi\end{align}$$

3. solve for $$\begin{align}y(x)\end{align}$$

1st condition of exactness
The 1st condition of exactness can be satisfied, if the equation is written in the form of eqn(1.2)

where,

$$\begin{align}p:=y'\end{align}$$

$$\begin{align}g(x,y,p) := {\phi}_{x}+{\phi}_{y}p\end{align}$$

$$\begin{align}f(x,y,p) := {\phi}_{p}(x,y,p)\end{align}$$

let's rearrange the eqn(1.1) in the form of eqn(1.2)

where,

$$\begin{align}g(x,y,p)= \underbrace{({x}^{2}-\mathrm{sin}x)}_{{\phi}_{y}}p+\underbrace{2xy}_{{\phi}_{x}}\end{align}$$

$$\begin{align}f(x,y,p)=\underbrace{\mathrm{cos}x}_{{\phi}_{p}}\end{align}$$

The given equation is satisfied the 1st condition of exactness.

2nd condition of exactness
There are two relations for the 2nd condition of exactness.


 * first relation


 * second relation

As both exactness conditions are satisfied, we can conclude that the eqn(1.1) is exact.

Find $$\begin{align}\phi\end{align}$$
As,

using definition of $$\begin{align}g(x,y,p)\end{align}$$

It should be the same to the $$\begin{align}g(x,y,p)\end{align}$$ in the eqn(1.3)

Assume,

Then,

Let's derivate eqn(1.13) with respect to y to find $$\begin{align}f(y)\end{align}$$

it should be the same to the $$\begin{align}{h}_{y}\end{align}$$ in eqn(1.11)

Then,

let's put it to the eqn(1.8)

by rearranging, we can get,

where, $$\begin{align}k={k}_{2}-{k}_{1}\end{align}$$

we can easily check wheter it is correct or not by find partial derivatives wrt x, y, p

using definition of $$\begin{align}F(x,y,y',y'')\end{align}$$

This is the same equation to the eqn(1.1). So we know that eqn(1.17) is correct.

Solve for $$\begin{align}y(x)\end{align}$$
Now, we have to solve N1_ODE_VC to find y(x). let's rearrange eqn(1.17)

let's find integrating factor i(x) first.

then, we can find y(x).