User:EGM6321.f12.team7.Zhou/R3

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Problem 1: Show that integration constant $$k_1$$ is unessesary
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Given: L1-ODE-VC
The equation is given by,

When the $$P(x)\neq 0$$, we have,

$$y'+\underbrace{\frac {P(x)}{Q(x)}}_{a_o(x)}y=\underbrace{\frac{R(x)}{P(x)}}_{b(x)}$$

Use IFM method, and suppose $$h(x,y)$$ is only a founction of $$x$$, the integraion factor is given by,

Find
Verify that,$$k_1$$ is unnecessary and there is only one integration factor

Solution

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Simplify the equation, we have,

So, for equation$$, we have,

With equation $$, the integration of y is given by,

Obviously, $$k_1$$ is unnecessary.

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Problem 2: Compare the result
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Given: The result in lecture note and King's Book
The result of general homogeous L1-ODE-VC is given by ,

The solution of King's book is given by,

Find
Show that the solution of lecture agrees with the King's solution

Solution

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The homogeous solution of King 's result is given by,

The particular solution of King's result is given by,

Follow the notation used in lecture, the homogeous solution of of equation is given by,

The particular solution of equation is given by,

Obviouly, $$y_p ,y_h $$are similar to $$y_p'. y_h'$$.

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Problem 3: Calculate the homogeous result
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Find:The homogenous solution
Calculate the homogenous solution.

Solution

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Follow the notation used in lecture,the solution of $$a_o(x)y+y'=0 $$ is given by,

Where $$h(x)$$ is integration factor, $$b(x)$$ is equal to zero.

So, the homogeous solution is given by,

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Problem 3: Find the conunterpart
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Given:the general class of N1-ODE-VC
A general class of N!-ODE-VC that are either exact of can be exact by IFM method is given by,

Where $$\bar b(x,y)$$ and $$\bar c(x,y)$$ are given by,

Find:Find the counterpart of the general class of N1-ODE-VC
Find the counterpart of general class of N1-ODE-VC which is either exact or can be exact by IFM method.

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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The equation is based on an assumption that intergration factor is only a founction of x, which is given by,

Where,

So ,the counterpart of the intergration factor is given by,

To satisfy the condition that $$h$$ is function of y, we can set the below forms,

So,for$$ M(x,y)$$, we have,

the $$k_1$$ should be a constant since the h is only a founction of y. The N(x,y) is given by,

Let's set the $$\int^y b(t)dt + k_1$$ equal to $$\bar b(y)$$, $$ \int^x c(t)dt + k_2{y}$$ equal to $$\bar c(x,y)$$, then we have the conterpart of general class of N!-ODE-VC, which is given by,

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