Laplace Transform

Directly verify that the solution obtained by transforming a time-domain differential equation satisfies the original problem.

Starting conditions

$D=\frac{x^2-2x+1}{x^2-3x+2}$: simplify by cancellation. For value comparisons, use $x\in\{0,3\}$.

Procedure

1. Simplify D by cancellation, and separately identify the x values that make the original denominator 0.
2. At $x\in\{0,3\}$, check that the values before and after cancellation agree. Explain why $x\in\{1,2\}$ cannot be checked in the same way.

Results to retain

• Expressions before and after cancellation, excluded values, and substitution results at the two points.

Starting conditions

Transform $f(t)=e^{-2t}$ for t>0, then take its inverse transform.

Procedure

1. Use sym.laplace_transform and inverse_laplace_transformlaplace and ilaplace to obtain the transform and inverse transform.
2. Compare the original and recovered function values at $t\in\{.5,1,2\}$.
3. Explain how the pole −2 of the transformed expression relates to the time-domain decay term $e^{-2t}$.

Results to retain

• A short table of F(s), the recovered function, and numerical differences.

Starting conditions

$\dot y+2y=1$; compare y(0)=0 with y(0)=1. The input is a unit step for t≥0.

Procedure

1. Write the Laplace transform of the derivative term and substitute the initial condition to obtain $(s+2)Y(s)=\frac1s+y(0)$.
2. Divide both sides of this equation by $s+2$ to obtain $Y(s)$. Substitute each initial condition, take the inverse transform, and plot both $y(t)$ over $0\le t\le4$ s together.
3. For each $y(t)$, calculate $\frac{dy}{dt}+2y-1$ and $y(0)-y_0$. Record both residuals for each initial condition and determine whether they are 0.

Results to retain

• Two response curves, a residual table, and a brief explanation of the initial condition's role.