University of Florida/Egm4313/s12.team8.dupre/R3.4

Problem Statement
Use the Basic Rule and the Sum Rule to show that the appropriate particular solution to $$\displaystyle y''-3y'+2y=4x^2-6x^5 $$      (4.1) is of the form $$\displaystyle y_{p}(x)=\sum_{j=0}^{n}c_{j}x^{j} $$ with n=5, i.e, $$\displaystyle y_{p}(x)=\sum_{j=0}^{5}c_{j}x^{j} $$.

Solution
The Basic rule and Sum rule allow us to choose our particular y forms from table 2.1 on page 82 in the Kreyszig Advanced Engineering Mathematics book. These rules state that, using your given $$\displaystyle r(x)$$, you can find what $$\displaystyle y_{p}(x) $$ you should choose to use. From (4.1), we know that: $$\displaystyle r(x)=4x^2-6x^5 $$     (4.2) Referring to table 2.1, and knowing that the Basic rule tells us that we can match up the form of our $$\displaystyle r(x)$$ to one in the table, and use the $$\displaystyle y_{p}(x) $$ accordingly, we find the following two equations: The $$\displaystyle y_{p}(x) $$ for $$\displaystyle 4x^{2} $$ is: $$\displaystyle k_{2}x^{2}+k_{1}x^{1}+k_{0} $$     (4.3) And the $$\displaystyle y_{p}(x) $$ for $$\displaystyle 6x^{5} $$ is: $$\displaystyle K_{5}x^{5}+K_{4}x^{4}+K_{3}x^{3}+K_{2}x^{2}+K_{1}x^{1}+K_{0} $$     (4.4) The sum rule tells us that we are able to add these equations, (4.3) and (4.4), to obtain a final solution for $$\displaystyle y_{p}(x) $$. The solution for this is as follows: $$\displaystyle k_{2}x^{2}+k_{1}x^{1}+k_{0} +K_{5}x^{5}+K_{4}x^{4}+K_{3}x^{3}+K_{2}x^{2}+K_{1}x^{1}+K_{0} $$ Adding similar variables gives us: $$\displaystyle K_{5}x^{5}+K_{4}x^{4}+K_{3}x^{3}+(K_{2}+k_{2})x^2+(K_{1}+k_{1})x^1+(K_{1}+k_{0}) $$    (4.5) Since these k and K values are just constants, we can set: $$\displaystyle K_{5}=c_{5} $$ $$\displaystyle K_{4}=c_{4} $$ $$\displaystyle K_{3}=c_{3} $$ $$\displaystyle (K_{2}+k_{2})=c_{2} $$ $$\displaystyle (K_{1}+k_{1})=c_{1} $$ $$\displaystyle (K_{1}+k_{0})=c_{0} $$ Our FINAL solution for the $$\displaystyle y_{p}(x) $$ of (4.1) is: $$\displaystyle y_{p}(x)=c_{5}x^{5}+c_{4}x^{4}+c_{3}x^{3}+c_{2}x^{2}+c_{1}x^{1}+c_{0} $$    (4.6) And, as was the original problem statement, equation (4.6) is of the form: $$\displaystyle y_{p}(x)=\sum_{j=0}^{5}c_{j}x^{j} $$