MATH202 - Ordinary Differential Equations & Laplace Transforms Flashcards

Question 1

First-Order Separable ODE

Accepted Answer

A first-order equation is **separable** if it can be written as $g(y) \, dy = f(x) \, dx$. Integrate both sides directly to solve: $\int g(y) \, dy = \int f(x) \, dx + C$.

Question 2

First-Order Linear ODE & Integrating Factor

Accepted Answer

Standard form: $y' + P(x)y = Q(x)$.

Solved using the **integrating factor** $I(x) = e^{\int P(x) \, dx}$. Multiplying both sides yields $\frac{d}{dx}[I(x)y] = I(x)Q(x)$, which integrates to:
$y(x) = \frac{1}{I(x)} \int I(x)Q(x) \, dx$

Question 3

Exact Differential Equations

Accepted Answer

An equation $M(x,y)dx + N(x,y)dy = 0$ is **exact** if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. The implicit solution is $\Psi(x,y) = C$, where $\frac{\partial \Psi}{\partial x} = M$ and $\frac{\partial \Psi}{\partial y} = N$.

Question 4

Homogeneous Linear ODE with Constant Coefficients

Accepted Answer

For $a y'' + b y' + c y = 0$, assume a trial solution $y = e^{rt}$. Substituting $y$ yields the **characteristic equation** $a r^2 + b r + c = 0$.

Question 5

Characteristic Equation: Distinct Real Roots

Accepted Answer

If $a r^2 + b r + c = 0$ yields two distinct real roots $r_1 \
eq r_2$, the general solution is:
$y(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t}$

Question 6

Characteristic Equation: Repeated Real Roots

Accepted Answer

If $a r^2 + b r + c = 0$ yields a repeated real root $r_1 = r_2 = r$, the general solution is:
$y(t) = c_1 e^{rt} + c_2 t e^{rt}$

Question 7

Characteristic Equation: Complex Conjugate Roots

Accepted Answer

If the characteristic roots are complex conjugates $r = \alpha \pm i\beta$, the general solution is:
$y(t) = e^{\alpha t} (c_1 \cos(\beta t) + c_2 \sin(\beta t))$

Question 8

The Wronskian & Linear Independence

Accepted Answer

The Wronskian of functions $y_1(t)$ and $y_2(t)$ is:
$W(y_1, y_2)(t) = y_1 y_2' - y_1' y_2 = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix}$
If $W \
eq 0$ on an interval, $\{y_1, y_2\}$ are linearly independent and form a fundamental set of solutions.

Question 9

Method of Undetermined Coefficients

Accepted Answer

A technique to find a particular solution $y_p(t)$ for non-homogeneous ODEs $a y'' + b y' + c y = g(t)$ when $g(t)$ contains polynomials, exponentials, or sines/cosines. Guess $y_p$ matching $g(t)$'s family, multiplying by $t^s$ if any term overlaps with the homogeneous solution.

Question 10

Method of Variation of Parameters

Accepted Answer

A general method for finding $y_p(t)$ for $y'' + p(t)y' + q(t)y = g(t)$ using fundamental solutions $y_1, y_2$:
$y_p(t) = -y_1(t) \int \frac{y_2(t)g(t)}{W(y_1,y_2)} dt + y_2(t) \int \frac{y_1(t)g(t)}{W(y_1,y_2)} dt$

Question 11

Cauchy-Euler Second-Order Equation

Accepted Answer

An equation of the form $a x^2 y'' + b x y' + c y = 0$. Substituting $y = x^r$ yields the characteristic equation $a r(r-1) + b r + c = 0$ (valid for $x > 0$).

Question 12

Definition of the Laplace Transform

Accepted Answer

The Laplace transform converts a time-domain function $f(t)$ ($t \ge 0$) into a frequency-domain function $F(s)$:
$\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt$

Question 13

Laplace Transform of First & Second Derivatives

Accepted Answer

• $\mathcal{L}\{f'(t)\} = s F(s) - f(0)$
• $\mathcal{L}\{f''(t)\} = s^2 F(s) - s f(0) - f'(0)$

Where $F(s) = \mathcal{L}\{f(t)\}$. Allows reducing ODEs to algebraic equations.

Question 14

Basic Laplace Transforms: Polynomials & Exponentials

Accepted Answer

• $\mathcal{L}\{1\} = \frac{1}{s}$
• $\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}$ (for $n \in \mathbb{N}$)
• $\mathcal{L}\{e^{at}\} = \frac{1}{s-a}$ (for $s > a$)

Question 15

Basic Laplace Transforms: Trigonometric Functions

Accepted Answer

• $\mathcal{L}\{\sin(\omega t)\} = \frac{\omega}{s^2 + \omega^2}$
• $\mathcal{L}\{\cos(\omega t)\} = \frac{s}{s^2 + \omega^2}$

Question 16

First Shifting Theorem (Frequency Shift)

Accepted Answer

Multiplying $f(t)$ by an exponential $e^{at}$ shifts the transform in the $s$-domain:
$\mathcal{L}\{e^{at} f(t)\} = F(s - a)$

Question 17

Heaviside Step Function & Second Shifting Theorem

Accepted Answer

The unit step function is $u(t-a) = \begin{cases} 0, & t < a \\ 1, & t \ge a \end{cases}$

**Second Shifting Theorem:**
$\mathcal{L}\{u(t-a) f(t-a)\} = e^{-as} F(s)$

Question 18

Dirac Delta Function & Its Laplace Transform

Accepted Answer

The Dirac delta function $\delta(t-a)$ models an instantaneous unit impulse at $t = a$.
Its Laplace transform is:
$\mathcal{L}\{\delta(t-a)\} = e^{-as}$ (for $a \ge 0$)

Question 19

Convolution Integral

Accepted Answer

The convolution of two functions $f(t)$ and $g(t)$ defined for $t \ge 0$ is:
$(f * g)(t) = \int_{0}^{t} f(\	au) g(t - \	au) \, d\	au$

Question 20

Convolution Theorem for Laplace Transforms

Accepted Answer

The Laplace transform of a convolution equals the product of individual transforms:
$\mathcal{L}\{(f * g)(t)\} = F(s) \cdot G(s)$
Inversely: $\mathcal{L}^{-1}\{F(s)G(s)\} = (f * g)(t)$.

Question 21

System of First-Order Linear ODEs (Eigenvalue Method)

Accepted Answer

For a homogeneous system $\mathbf{x}' = A\mathbf{x}$, if $A$ is an $n \	imes n$ matrix with $n$ distinct eigenvalues $\lambda_i$ and eigenvectors $\mathbf{v}_i$, the general solution is:
$\mathbf{x}(t) = c_1 \mathbf{v}_1 e^{\lambda_1 t} + c_2 \mathbf{v}_2 e^{\lambda_2 t} + \dots + c_n \mathbf{v}_n e^{\lambda_n t}$

MATH202 - Ordinary Differential Equations & Laplace Transforms

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