Laplace Transform

Learning objectives

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

Checking that simplification preserves values

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.

import sympy as sym
x=sym.symbols('x')
D=(x**2-2*x+1)/(x**2-3*x+2)
print('약분:',sym.cancel(D))
print('원래 분모의 근:',sym.solve(x**2-3*x+2,x))
# TODO: Compare values at x=0,3 and explain the two excluded points.

Record results and self-check

Original expression before cancellation:

Expression after cancellation:

Excluded x values:

Value difference at the two points:

Points to check

• Did you identify both values that make the original denominator 0?
• Did you avoid substituting $x=1$ into the original expression based only on its simplified form?

Laplace transform and inverse transform

Starting conditions

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

Procedure

1. Use sym.laplace_transform and inverse_laplace_transform 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.

import sympy as sym
t=sym.symbols('t',positive=True);s=sym.symbols('s')
f=sym.exp(-2*t)
F=sym.laplace_transform(f,t,s,noconds=True)
restored=sym.inverse_laplace_transform(F,s,t)
print(F,restored)
# TODO: Check the differences at the three specified times.

Record results and self-check

Function:

F(s):

Inverse transform:

Maximum numerical error:

Points to check

• Did you record the positive assumption for t and the initial condition?
• At the same t>0, is the numerical error between the original function and its inverse transform at most 1e-8?

Differential equations with initial conditions

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.

import sympy as sym
t=sym.symbols('t',positive=True); s=sym.symbols('s')
for y0 in [0,1]:
    Y=(1/s+y0)/(s+2)
    y=sym.inverse_laplace_transform(Y,s,t)
    print(y0,y)
# TODO: Add residual checks and plots.

Record results and self-check

y(0):

Y(s):

Initial-value residual:

ODE residual:

Steady-state value:

Points to check

• Did the input and initial condition enter as separate terms?
• Is the symbolic residual 0, and do both responses have the same steady-state value?