User:Egm6321.f11.team1.colsonfe/HW5

= R*5.1 2nd Exactness Condition For n=2 Case of Nn-ODE (Method 2) =

Given
Equivalent form of 2nd Exactness Condition for N2-ODEs :


 * {| style="width:100%" border="0" align="left"

$$
 * $$g_0-\frac{dg_1}{dx}+\frac{d^2g_2}{dx^2}=0$$
 * $$\displaystyle (Equation\;4.7.1)
 * $$\displaystyle (Equation\;4.7.1)
 * }
 * }

Coefficient equalities, located in previous reference above.


 * $$\displaystyle g_0=f_yq+g_y$$


 * $$\displaystyle g_1=f_pq+g_p$$


 * $$\displaystyle g_2=f$$

Note also that $$\displaystyle p(x):=y'(x)$$ and $$\displaystyle q(x):=y''(x).$$

== Find ==

Show the following equality is true:


 * $$\displaystyle f_{xx}+2pf_{xy}+p^2f_{yy}-g_{xp}-pg_{yp}+g_y+\left( f_{xp}+pf_{yp}+2f_y-g_{pp}\right)q=0$$

Solution
Applying the chain rule to $$\frac{dg_1}{dx}$$ yields the following:

$$\frac{dg_1}{dx}=\frac{\partial g_1}{\partial x}+\frac{\partial g_1}{\partial y}\frac{dy}{dx}+\frac{\partial g_1}{\partial y^{(1)}}\frac{dy^{(1)}}{dx}$$

We know what $$\displaystyle g_1$$ is defined to be, we can substitute in $$\displaystyle p(x):=y'(x)$$ and $$\displaystyle q(x):=y''(x)$$ and carry out the derivatives:

$$\frac{dg_1}{dx}=\frac{\partial g_1}{\partial x}+\frac{\partial g_1}{\partial y}p+\frac{\partial g_1}{\partial p}q$$

$$\frac{dg_1}{dx}=\left(f_{xp}q+g_{xp}\right)+\left(f_{yp}q+g_{yp}\right)p+\left(f_{pp}+g_{pp}\right)q$$

We can apply the same approach to $$\frac{d^2g_2}{dx^2}$$ :

$$\frac{d^2g_2}{dx^2}=\frac{\partial^2 g_2}{\partial x^2}+\frac{\partial^2 g_2}{\partial y\partial x}\frac{dy}{dx}+\frac{\partial^2 g_2}{\partial x\partial y}\frac{dy}{dx}+\frac{\partial^2 g_2}{\partial x\partial y^{(1)}}\frac{dy^{(1)}}{dx}+\frac{\partial^2 g_2}{\partial y^{(1)}\partial x}\frac{dy^{(1)}}{dx}+\frac{\partial^2 g_2}{\partial^2 y}\frac{dy^2}{dx^2}$$

$$\frac{d^2g_2}{dx^2}=f_{xx}+f_{yx}p+f_{xy}p+f_{xp}q+f_{px}q+f_{yy}p^2$$

Adding $$\displaystyle g_0 $$, $$\displaystyle \frac{dg_1}{dx}$$ , and $$\displaystyle \frac{d^2g_2}{dx^2}$$ according to Equation 4.7.1 yields our final solution:

$$ g_0-\frac{dg_1}{dx}+\frac{d^2g_2}{dx^2}=\left(f_yq+g_y\right)-\left(f_{xp}q+g_{xp}+f_{yp}pq+g_{yp}p+g_{pp}q\right)+\left(f_{xx}+f_{yy}p^2+2f_{xy}p+2f_{xp}q\right)=0$$

= References =