User:EGM6321.f12.team7.Zhou/R6

Problem 4: Method of Variation of Parameter
Report problem 6.4 from section 32.

Given: The Characteristic Equation
The characteristic equation is given by,

Where $$ \lambda = 5$$ is given. The Euler-L2-ODE-VC is given by,

The Euler-L2-ODE-CC is given by,

Find: The first and complete solution of equations
1. Find $$ a_2,a_1, a_0$$ such that the characteristic function is ($$).

2. Find the first homogeneous solution.

3. Find the complete solution.

4. Find the second homogeneous solution.

5. Repeat the above steps to equation

Solution

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 * On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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Figure out the coefficient of the equations
So we use the trial solution, which is given by ,

So the derivation of y is given by,

So drive the equation($$)into equation($$), we have,

The characteristic equation is given by,

According to the linear-independence, we can have,

The Euler-L2-ODE-VC then changes into,

Find out the first homogenous solution
The first homogenous solution is given by,

Figure out the complete solution
Use the Variation of Parameters, we assume that the complete solution is given by,

So drive equation ($$)into equation ($$), we have,

Suppose Z=U', we can reduce the order of equation, which is given by,

So now the equation is L1-ODE-VC.When $$ x^5 \neq 0$$, both sides are divided by $$x^5$$, we will have,

Use Integration Factor Method, we have,

So the solution of L1-ODE-VC is given by,

So according to equation ($$), we will have,

So the complete solution is given by,

Figure out the second homogeneous solution
The second homogeneous solution is given by,

Solve the other Euler L2-ODE-CC
So use trial solution, we have,

Drive equation i ($$)into equation ($$), we have,

According to equation and property of linear independence, we have,

So the Rulor-ODE-CC turns into,

Use Method of Variation of Parameters, we assume that the complete solution is given by,

So drive equation ($$) into equation ($$), we have,

Suppose $$ Z=U'$$, we can reduce the order of equation,

The Integration Factor is given by,

So the solution of L1-ODE-CC is given by,

The solution of U is given by,

So the complete solution is given by,

Problem 7: Show Equivalence between two expressions
Report problem 6.7 from section 35.

Given: Two expressions of paticular solution of y
From lecture note, the L2-ODE-VC is given by,

The particular solution is given by

Where the $$ u_1(x) $$ is the 1st homogeneous solution and h(x) is integration factor.

In King's book, for the same equation, the particular solution is given by,

Where $$ u_1, u_2$$ are the 1st and 2nd homogeneous solution, and W(s) is wroskian function, which is given by,

Some choices of variation of parameters,

Find: equivalence of two expressions and the feasibility
1. Show the equivalence of two expressions of particular solution 2 Test the feasibility of the other choices of variation of parameters

Solution

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

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

Show the equivelance
According to lecture note, the expression of the 2nd homogeneous solution is given by,

So we can derive the below expression,

So the derivative of equation can be expressed by,

The expression of King's book can be reorganized into below,

So for equation($$), use integrations by parts , we will have,

So the equivalence has been prooved.

check the feasibility
For the equation, we just consider the$$'+'$$, since the other situation is the same. So we have,

So the equation turns into.

So we just circle back to the original equation and nothing can be done to reduce the order.

For the second choice, we have,

So drive equation back to equation, the coefficient of U is given by,

Obviously, there is no way to make it equal to zero, so we can not reduce the order of equation.

For the third choice,we have

We can see that this choice has make the order of $$U$$ even higher, which strongly implies that the equation cannot be solved by this choice.