7.5 Workshop Exercises 5

MATH102 Integration 2016 Workshop Exercise 5

For more worked examples, see Gilbert and Jordan Guide to Mathematical Methods, pages 153–164.

This week’s exercises cover: second order differential equations, constant coefficients (W5.1-5.3); Laplace transforms (W5.4-5.5).

W5.1. Find the general solution of each of the following linear inhomogeneous equations

W5.2. Solve the following initial value problems:

a) $\frac{d^{2}y}{dx^{2}}-\frac{dy}{dx}-2y=e^{x}$, $y(0)=2$, $y^{\prime}(0)=1$.

b) $\frac{d^{2}y}{dx^{2}}-\frac{dy}{dx}-2y=e^{-x}$, $y(0)=2$, $y^{\prime}(0)=1$.

c) $\frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}+y=e^{x}$, $y(0)=2$, $y^{\prime}(0)=1$.

W5.3. Find the general solution of the equation $\frac{d^{2}y}{dx^{2}}-6\frac{dy}{dx}+9y=4e^{3x}$.

Material on Laplace transforms is non-examinable, and is included for interest only.

W5.4. Find the Laplace transforms of the following functions:

You may use the table of transforms on slide 6.44 in the notes, as well as standard properties of Laplace transforms such as linearity (Thm. 6.40) and the shift formula.

W5.5. i) Obtain the Laplace transform of the function $f(x)=e^{x}\cosh x$, using the table on slide 6.44 and linearity.

ii) Use (i) and the Laplace transform method (see Examples 6.49 and 6.50 in the notes) to find the solution to the initial value problem: