User:Egm6321.f12.team07.liu/report4

=R*4.4=

Problem 4: Solve certain L2-ODE-VC
Report problem 4 from.

Given: A L2-ODE-VC
The linear 2nd-order ordinary differential equation with varying coefficients(L2-ODE-VC):

Find: Show exactness and solve through integration

 * 1) Show the equation is exact
 * 2) Find first integration $$\phi$$
 * 3) Solve for $$y(x)$$

Solution of the L2-ODE-VC

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

This solution was produced without referring to previous solutions. 1, Show exactness:
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * }

First write ($$) into:

Obiviously, ($$) represents that the L2-ODE-VC satisfies the 1st exactness conditin.

Then, as for the 2nd exactness condition, we need to check if ($$) satisfies the following expressions:

we have: $$LHS1=RHS1=-cosx$$, $$LHS2=RHS2=0$$

Thus, ($$)satisfies the 1st and 2nd exactness conditions i.e. it is exact.

2, Find $$\phi$$

As for ($$), we know that there exist a $$\phi$$ which satisfies:

According to ($$), we have:

Moreover, we obtain:

{{NumBlk|:|$$\left\{\begin{matrix} \phi _y=h_y\\ \phi _x=-psinx+h_x\end{matrix}\right.$$|$$}}

Institute ($$) into ($$) and make integration, we obtain:

Where $$k_1$$ is a constant.

Finally, institute ($$) into ($$), we have:

3, Solve for $$y(x)$$

Then our problem to solve $$y(x)$$ turns to solve equation:

i.e.

It is easy to find that: $$M_y\neq N_x$$

Thus ($$) is not exact, we consider using IFM:

Introduce $$h(x)$$ such that:

to render it exact, we have: $$\bar{M}_y=\bar{N}_x$$

More specificly:

Through integration, we obtain:

Moreover to get $$y(x)$$, we need to find the integration $$\bar{\phi}$$ of ($$):

we have:

According to ($$), we obtain:

Institute ($$) into ($$), we get:

Then we get:

i.e.

Plug ($$) back into ($$), we obtain the result:

Where $$h(x)$$ is decided by ($$)