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Calculus
Differential Equations Introduction
First-order ODEs and exponential growth
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A differential equation is an equation that connects an unknown function to its derivatives, so it describes how a quantity changes rather than just what its value is. First-order ordinary differential equations involve one independent variable and the first derivative of the unknown function. They are central in physics, biology, economics, and engineering because many systems are defined by rates of change. A slope field gives a visual map of these rates and shows how many possible solution curves can flow through the same equation.
Key Facts
- A first-order ODE has the general form dy/dx = f(x, y).
- An initial value problem combines a differential equation with a starting value, such as dy/dx = f(x, y), y(x0) = y0.
- A separable differential equation can be written as dy/dx = g(x)h(y).
- Separation of variables rewrites dy/dx = g(x)h(y) as dy/h(y) = g(x) dx, then integrates both sides.
- Exponential growth and decay follow dy/dt = ky, with solution y = Ce^(kt).
- If k > 0, y = Ce^(kt) grows exponentially; if k < 0, it decays exponentially.
Vocabulary
- Ordinary differential equation
- An equation involving an unknown function of one independent variable and one or more of its derivatives.
- First-order equation
- A differential equation whose highest derivative is the first derivative.
- Slope field
- A diagram that shows the slope dy/dx at many points in the plane for a differential equation.
- Separation of variables
- A method for solving certain differential equations by moving all terms involving y to one side and all terms involving x to the other side.
- Initial condition
- A specified value of the unknown function at a particular input, used to select one solution from a family of solutions.
Common Mistakes to Avoid
- Forgetting the constant of integration after integrating both sides is wrong because the constant represents the family of possible solution curves.
- Treating every first-order ODE as separable is wrong because separation only works when the equation can be rearranged into y terms on one side and x terms on the other.
- Dividing by a variable expression without checking when it equals zero is wrong because it may discard constant or equilibrium solutions.
- Using the sign of k incorrectly in dy/dt = ky is wrong because positive k gives growth while negative k gives decay.
Practice Questions
- 1 Solve dy/dx = 3y with initial condition y(0) = 4.
- 2 Solve dy/dx = 2xy with initial condition y(0) = 5.
- 3 A slope field for dy/dx = y shows horizontal line segments along y = 0 and steeper positive slopes for larger positive y. Explain what this suggests about equilibrium solutions and long-term behavior for solutions starting above y = 0.