User:Egm4313.s12.team8.lucas/R2.2

=R2.2=

Problem Statement
Find and plot the solution for the L2-ODE-CC Eqn. 4 ($$\displaystyle y''-10y'+25y=r(x)$$) on p. 5-5 in the lecture notes. The initial conditions are $$\displaystyle y(0)=1$$,  $$\displaystyle y'(0)=0$$ with no excitation ($$\displaystyle r(x)=0$$)

Solution
Also because $$\displaystyle r(x)=0$$, the ODE becomes:

Which can be written as the characteristic equation:

Factoring the characteristic equation gives:

Which solves giving a double real root:

The double real root means the solution will have the form:

The first homogeneous solution is:

The second homogeneous solution for a double real root takes the form:

The final homogeneous solution is:

Applying the initial condition $$\displaystyle y(0)=1$$ to the solution gives:

The updated solution is: Differentiting the solution gives: Applying the second initial condition, $$\displaystyle y'(0)=0$$ gives:

The final solution is:



The graph above is a MATLAB plot of the solution to the ODE: $$\displaystyle y(x)=e^{5x}-5xe^{5x}$$

Contributions
Solved and typed by: Egm4313.s12.team8.lucas 04:35, 8 February 2012 (UTC)

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