Introduction to Differential Equations
This worksheet introduces Grade 10 students to basic concepts and methods for solving differential equations, including separation of variables and initial value problems.
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Introduction to Differential Equations
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Read each question carefully and provide your answers in the space provided. Show all your work for full credit.
1. A differential equation is an equation that involves a function and its .
2. The of a differential equation is the order of the highest derivative in the equation.
3. A solution to a differential equation is a function that satisfies the equation when into it.
4. Which of the following is an example of a first-order differential equation?
$$y'' + y = 0$$
$$\frac{dy}{dx} = x + y$$
$$\frac{d^2y}{dx^2} + x\frac{dy}{dx} = 0$$
$$y''' - y' = 5$$
5. Which method is commonly used to solve differential equations where the variables can be isolated on opposite sides of the equation?
Integration by Parts
Separation of Variables
L'Hôpital's Rule
Partial Fractions
6. Verify that $$y = e^{2x}$$ is a solution to the differential equation $$\frac{dy}{dx} - 2y = 0$$.
7. Solve the differential equation $$\frac{dy}{dx} = 3x^2$$ with the initial condition $$y(0) = 1$$.
8. An initial value problem (IVP) for a differential equation includes the differential equation itself and one or more initial conditions.
True
False
9. Solve the following differential equation using separation of variables: $$\frac{dy}{dx} = \frac{x}{y}$$.
10. Consider a population of bacteria that grows at a rate proportional to its current size. If the population doubles every hour and initially there are 100 bacteria, write down and solve the differential equation that models this growth.