User:Egm4313.sp12.team4.diroma

test http://en.wikiversity.org/wiki/User:Egm4313.sp12.team4.diroma/report1

$$ function \!$$

=Problem 1=

Given the two roots:

$$\displaystyle \lambda_1 = -2 \! $$

$$\displaystyle \lambda_2 = 5 \! $$

and initial conditions:

$$\displaystyle y(0) = 1 \!$$

$$\displaystyle y'(0) = 0 \!$$

Find the non-homogeneous L2-ODE-CC in standard form and the solution in terms of the initial conditions and the general excitation $$ r(x) \! $$


 * Consider $$\displaystyle r(x) = 0 \! $$


 * Plot the solution

Solution
Characteristic Equation:

$$(\lambda-\lambda_1)(\lambda-\lambda_2) \!$$

Using the Given values for $$ \lambda_1 \! $$ and $$ \lambda_2 \! $$ we can write:

$$ (\lambda+2)(\lambda-5) = 0  \rightarrow   \lambda^2-3\lambda-10 = 0  $$

Non-homogeneous L2-ODE-CC:

$$ y''-3y'-10y = r(x) \! $$

The homogeneous solution is given by:

$$ y_h = c_1 e^{-2x} + c_2 e^{5x} \! $$

The overall solution is given by:

$$ y(x) = y_h + y_p \! $$;           where $$ y_h $$ is the homogeneous solution and $$ y_p $$ is the particular solution.

When the excitation factor $$ r(x) \! $$ is zero (as in this case), $$ y_p $$ is zero, so we can use the given initial conditions to find constants $$ c_1 \!$$ and $$ c_2 \!$$.

To do this we must first take the derivative $$ \frac{dy}{dx} $$ of $$ y_h \!$$.

Doing so, we find $$ y'_h = 0 = -2c_1+5c_2 \!$$

Multiplying $$ y_h \!$$ by 2 and adding to $$ y'_h \!$$ we can solve for $$ c_2 \!$$

$$ c_2 = \frac{2}{7} \!$$

Then multiplying $$ y_h $$ by -5 and adding to $$ y'_h $$ we can solve for $$ c_1 $$

$$ c_1 = \frac{5}{7} \!$$

Substituting in our new values for $$ c_1 \!$$ and $$ c_2 \!$$ we find the final solution to be:


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


 * style="width:95%" |
 * style="width:95%" |
 * }

To obtain three non-standard (non homogeneous) L2-ODE-CC that correspond to the same roots of our characteristic equation, we can simply multiply our characteristic equation by any non-zero constant. So choosing 2,3 and 4 as our constants we obtain:


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


 * style="width:95%" |
 * style="width:95%" |
 * 
 * }


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


 * style="width:95%" |
 * style="width:95%" |
 * 
 * }


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


 * style="width:95%" |
 * style="width:95%" |
 * 
 * }

These will all yield the same roots; $$ \lambda_1 \! $$ and $$ \lambda_2 \! $$

Author
Solved and typed by - Egm4313.sp12.team4.diroma 04:23, 8 February 2012 (UTC) Reviewed By - Team 4

Edited by - Egm4313.sp12.team4.diroma 04:23, 8 February 2012 (UTC)