What determines the general solution of a homogeneous second-order ODE?
Nature of the roots
The general solution for complex conjugate roots `α ± βi` is `y = e^(αx)(A*cos(βx) + B*sin(βx))`, where `α` represents the real part and `β` represents the imaginary part.
Ordinary differential equations involve only ordinary derivatives.
True
Match the type of differential equation with its application:
Ordinary Differential Equations (ODEs) ↔️ Population growth
What is the order of the differential equation ²y/dx² + dy/dx + y = x?
2
The degree of a differential equation is the highest power of the highest derivative.
True
Steps to solve a first-order differential equation using separation of variables:
1️⃣ Separate the variables
2️⃣ Integrate both sides
3️⃣ Solve for the dependent variable
A differential equation is used to model real-world phenomena such as population growth, radioactive decay, and fluid dynamics
What is an example of a Partial Differential Equation (PDE)?
\frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2}</latex>
What is an example of a real-world system modeled by a Partial Differential Equation (PDE)?
Heat diffusion
What does an ordinary differential equation (ODE) involve?
Single variable and ordinary derivatives
What is the purpose of a differential equation?
Model real-world phenomena
The degree of a differential equation is the highest power of its dependent variable and derivatives.
True
What is the first step in solving a first-order differential equation using the separation of variables method?
Separate the variables
What does the order of a differential equation refer to?
Highest derivative present
In the equation `dy/dx + 2y = x`, the order is 1
The separation of variables method involves arranging terms with the dependent variable and its derivative on one side and terms with the independent variable on the other.
True
The separation of variables method is not applicable if the variables cannot be separated
Steps to solve a homogeneous differential equation
1️⃣ Identify the equation as homogeneous
2️⃣ Substitute a new variable
3️⃣ Separate the variables
4️⃣ Integrate both sides
5️⃣ Solve for the original variable
The integrating factor `μ(x)` is defined as e(∫P(x)dx) where `P(x)` is from the equation `dy/dx + P(x)y = Q(x)`.
True
For a second-order homogeneous ODE, the characteristic equation is obtained by substituting y = e^(mx) into the differential equation.
What is the general form of a second-order differential equation?
a(d2y/dx2)+b(dy/dx)+cy=0
The general solution for distinct real roots `m1` and `m2` is `y = A*e^(m1x) + B*e^(m2x)`.
True
Second-order ODEs are used in modeling physical phenomena like simple harmonic motion.
True
What type of derivatives do partial differential equations involve?