User:EGM6321.f10.team6.yoshioka/HW2

Required
Find the function of $$f\left( y' \right)$$ such that there is no analytical solution to $$f\left( y' \right)=-\frac{M\left( x,y \right)}{N\left( x,y \right)}$$.

Solution
The solution of the problem is equivalent to find a N1_ODE that does not satisfy the 1st condition of exactness. In order to a N1_ODE to be exact, it must be in the form of:

The function chosen to test the first condition of exactness is:

From Equation 5.3 we can identify the functions, $$M\left( x,y \right)$$, $$N\left( x,y \right)$$ and $$f\left( y' \right)$$ as:

If we can rearrange Equation 5.3 to fit in the form of the Equation 5.2, than the 1st condition of exactness, if not we found Non-Exact N1_ODE.

Doing some algebra with Equation 5.3 to arrange in the fashion of $$f\left( y' \right)=-\frac{M\left( x,y \right)}{N\left( x,y \right)}$$:

Equation 5.7 does not fit in the same arrangement of Equation 5.2, since we got $$y'+\cos y'$$ on the right hand side of the equation instead of only $$y'$$. Therefore there is no analytical solution to the function chosen and also proves that Equation 5.3 is a Non Exact N1_ODE.

Given
The Euler Integrating Factor Method may be applied to homogeneous differential equations when the equations meet the first condition of exactness, but do not meet the second condition of exactness. That is to say, those equations that can me made to fit the form shown in equation 7.1, but do not meet the criteria shown in equation 7.2 can be made exact using the Euler method. One of the benefits of the Integrating Factor Method is that it can be applied to solve non-homogenous differential equations.

Required
(i) Solve the following non-homogeneous differential equation using the Integrating Factor Method:

(ii) Assume that $${{a}_{1\left( x \right)}}\ne 0$$ for any $$x$$ and find an expression for $${{y}_{\left( x \right)}}$$ in terms of $${{a}_{0}}$$, $${{a}_{1}}$$, $${b}$$ for the following form:

(iii) Apply the integrating factor method to the following equation:

Explanation of Process
The Integrating Factor Method assumes that some function $${{h}_{\left( x,y \right)}}$$ can be found to make a differential equation exact when it meets the first criterion of exactness (equation 7.1) but not the second (equation 7.2). This factor is multiplied by the original equation to yield equation 7.6.

Applying the second condition of exactness and using the product rule to differentiate each term we obtain:

If we assume that the partial derivative of h with respect to y is zero, then h becomes a function of x only and we are left with the following:

Which can be rearranged to obtain:

Dividing by h and rearranging terms we obtain:

Integrating both sides with respect to x we obtain the following function we can use to obtain h:

If we return to our original non-exact equation given in equation 7.1 we can apply the integrating factor to obtain the following equation:

If we recall that $$h\left( {y}'+{{a}_{0}}y \right)=hb$$, we can rearrange the above to obtain the following:

The reader might recognize that the left side of equation 7.13 is really nothing more than $${{\left( hy \right)}^{\prime }}$$. Substituting into 7.13 we obtain:

After integrating we are left with the solution to y.

Solution to Part I
From equation 7.3 we obtain the following:

Substituting into equation 7.11 we obtain an expression for h:

Combing the result from equation 7.17 with equation 7.15 we obtain the solution to y.

Rearranging the terms:

Solving the first Integral

Using the substitution method

Substituting in the Euqation 7.? we get

And here I hit the wall, I have no idea how to solve the second integral...

Solution to Part II
From equation 7.3 we obtain the following:

Substituting into equation 7.11 we obtain an expression for h:

Combing the result from equation 7.20 with equation 7.15 we obtain the solution to y.

Solution to Part III
First we need to rearranging Equation 7.5 to be in the form of:

Rearranging Equation 7.5 we get:

Calculating $${{M}_{y}}$$, $$N$$ and $${{N}_{x}}$$:

Substituting into equation 7.11 we obtain an expression for h:

From equation 7.21 we have a general solution in terms of $${{a}_{0}}$$, $${{a}_{1}}$$, $${b}$$. Substituting the terms given in the problem statement in equation 7.5 we obtain:

Using the substitution method to solve the integral

Substituting Equation 7.24 into 7.15:

Using the substitution method to solve the integral

From Equation 7.27 we get the final answer for $$y(x)$$