User:Egm6321.f12.team07.liu/report6

=R*6.5=

Problem 5: Find the particular solution using method of variation of parameters
Report problem 5 from.

Given: A non-homogeneous L1-ODE-VC
The linear 1st-order ordinary differential equation with varying coefficients(L1-ODE-VC):

The homogeneous solution to ($$):

Find: The particular solution to the non-homogeneous L1-ODE-VC
Find the particular solution: $$y_P(x)$$

Solution: Using method of variation of parameters

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

On our honors, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions. Consider the following trial solution:
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * }

where $$A(x)$$ is the unknown to be found.

Instituting ($$) into ($$) yeilds:

i.e.

Solve the above equation, we have:

where $$C$$ is an integration constant.

Thus, the solution to ($$) works out to be:

From the solution above, the particular solution could be found:

where $$y_H(x)=exp[-\int_{}^{x}a_0(s)ds]$$ as ($$) already shown.

=R*6.12=

Problem 12: Find the final solution by variation of parameters
Report problem 12 from.

Given: A non-homogeneous Legendre equation and the 1st homogeneous solution
The non-homogeneous Legendre equation(L2-ODE-VC)

Given the 1st homogeneous solution being

Find: The final solution to the non-homogeneous Legendre equation
Find the final solution $$y(x)$$

Solution: Using method of variation of parameters

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

On our honors, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions. Consider the following trial solution
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * }

where $$A(x)$$ is the unknown to be found.

Instituting ($$) into ($$) yeilds

Rearrange ($$)

Assume that

then ($$) turns to be

Note that ($$) is a L1-ODE-VC which could be solved directly or by IFM.

Noticing that

{{NumBlk|:|$$ \left\{\begin{matrix} \phi_{Bx}=1-3x^2 \\ \phi_{xB}=2(1-2x^2) \end{matrix}\right.\Rightarrow \phi_{Bx}\neq \phi_{xB}$$|$$}}

i.e. ($$) is not exact and IFM should be used to solve it

Multiply ($$) with certain integration factor $$h(x)$$

Recall the 2nd exactness condition $$\bar{\phi}_{Bx}=\bar{\phi}_{xB}$$, we have:

i.e.

where $$k_1$$ is an integrating constant.

Moreover we could get the solution to ($$)

where $$k_2$$ is an integrating constant.

Integrating ($$) yeilds

where $$k_3$$ is another integrating constant.

Plug ($$) back into ($$), we get the final solution to ($$):

where $$C_1$$ and $$C_2$$ are two arbitrary constants.