User:Egm6321.f11.team4.allen/HW1

= Problem 1.8 - Find the Integration Constants in Terms of the Initial Conditions =

From [[media:Pea1.f11.mtg6.djvu|Mtg 6-5]]

Given
The general structure of Linear 2nd-Order ODEs with Varying Coefficients (L2-ODE-VC) is

$$\displaystyle P(x)y''+Q(x)y'+R(x)y=F(x). $$

The general solution can be written as

$$\displaystyle y(x)=C_1y_{H1}(x)+C_2y_{H2}(x)+y_P(x). $$

Find
Find the integration constants, $$\displaystyle C_1$$ and $$\displaystyle C_2$$, in terms of the following initial conditions:

$$\displaystyle y(a)=\alpha$$ and $$\displaystyle y'(b)=\beta$$.

Solution
First, we must differentiate the solution. This yields

$$\displaystyle y'(x)=C_1y'_{H1}(x)+C_2y'_{H2}(x)+y'_P(x)$$.

Next, we set up the two equations in order to solve for the two constants.

$$\displaystyle y(a)=\alpha=C_1y_{H1}(a)+C_2y_{H2}(a)+y_P(a)$$ (Eq. 1)

and

$$\displaystyle y'(a)=\beta=C_1y'_{H1}(x)+C_2y'_{H2}(a)+y'_P(a)$$. (Eq. 2)

Then, multiply Eq. 1 by $$\displaystyle y'_{H2}(a)$$ and Eq. 2 by $$\displaystyle y_{H2}(a)$$ which yields

$$\displaystyle y'_{H2}(a)\alpha=y'_{H2}(a)C_1y_{H1}(a)+y'_{H2}(a)C_2y_{H2}(a)+y'_{H2}(a)y_P(a)$$

and

$$\displaystyle y_{H2}(a)\beta=y_{H2}(a)C_1y'_{H1}(x)+y_{H2}(a)C_2y'_{H2}(a)+y_{H2}(a)y'_P(a)$$.

Solving the two equations for $$\displaystyle C_1$$ yields

$$\displaystyle C_1=\frac{y'_{H2}(a)\alpha-y_{H2}(a)\beta+y_{H2}(a)y'_P(a)-y'_{H2}(a)y_P(a)}{y'_{H2}(a)y_{H1}(a)-y_{H2}(a)y'_{H1}(a)}$$.

Similarly, multiply Eq. 1 by $$\displaystyle y'_{H1}(a)$$ and Eq. 2 by $$\displaystyle y_{H1}(a)$$ which yields

$$\displaystyle y'_{H1}(a)\alpha=y'_{H1}(a)C_1y_{H1}(a)+y'_{H1}(a)C_2y_{H2}(a)+y'_{H1}(a)y_P(a)$$

and

$$\displaystyle y_{H1}(a)\beta=y_{H1}(a)C_1y'_{H1}(x)+y_{H1}(a)C_2y'_{H2}(a)+y_{H1}(a)y'_P(a)$$.

Solving the two equations for $$\displaystyle C_2$$ yields

$$\displaystyle C_2=\frac{y'_{H1}(a)\alpha-y_{H1}(a)\beta+y_{H1}(a)y'_P(a)-y'_{H1}(a)y_P(a)}{y'_{H1}(a)y_{H2}(a)-y_{H1}(a)y'_{H2}(a)}$$.

Author
Contributed by Allen