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Calculus I (Differential & intro Integral) Unit 7: Intro differential equations

A differential equation relates a function to its own derivative. This unit covers solving the simplest kind and reading them as pictures.

Covers: Separable equations, slope fields, exponential growth & decay

Study guide

A differential equation relates a function to its derivatives, like dy/dx = 2xy. A solution is a function that makes the equation true; to verify one, substitute it and its derivatives into the equation. The general solution includes a constant C, and an initial condition such as y(0) = 3 picks out the particular solution.

A slope field draws short segments showing dy/dx at many points. You can sketch solution curves by following the segments, and you can match a slope field to its equation by checking simple cases: where are slopes zero, positive or negative, and do they depend only on x, only on y, or both?

Separable equations can be written dy/dx = g(x)h(y). Separate the variables so all y's are on one side with dy and all x's on the other with dx: (1/h(y)) dy = g(x) dx. Integrate both sides, add the constant C right away, use the initial condition to find C, and then solve for y if possible.

The most important example is exponential growth and decay: dy/dt = ky, which says the rate of change is proportional to the amount. Its solution is y = y₀e^(kt). If k > 0 the quantity grows; if k < 0 it decays. This models populations, compound interest, radioactive decay, and cooling (applied to the temperature difference).

To find d²y/dx² from a differential equation, differentiate the formula for dy/dx with respect to x (using implicit differentiation on y-terms), replace each dy/dx with its formula, then evaluate. The sign of the result gives the solution's concavity, and so tells whether a tangent-line estimate is too high or too low. To estimate a solution near a known point (x₀, y₀), plug the point into the differential equation to get the slope m, then use y ≈ y₀ + m(x − x₀).

Newton's law of cooling or warming, dT/dt = k(T − A), has the solution T = A + (T₀ − A)e^(kt), with k < 0. The temperature approaches the surrounding temperature A over time. Integrating (1/y) dy gives ln|y|. Use the sign of the initial value to remove the absolute value, and check that the final formula is defined on an interval containing the initial x-value.

Constant solutions (equilibria) occur at y-values where dy/dx = 0 for every x. In a slope field they show up as horizontal rows of segments that other solution curves never cross.

Key ideas

• Check a proposed solution by plugging it (and its derivative) into the equation.
• Separable equations: get all y-terms with dy and all x-terms with dx, integrate both sides, add C, then use the initial condition.
• A slope field draws the slope dy/dx as a short segment at each point. Solution curves follow the segments.
• dy/dt = ky means the rate is proportional to the amount; the solution is exponential.

Formulas and rules

Common mistakes

• Adding C after solving for y instead of right after integrating.
• Losing the initial condition step and leaving C unknown.

Words to know

Separable differential equation

dy/dx = g(x)h(y): separate, integrate both sides, then use the initial condition.

Solution to dy/dt = ky

y = y₀·eᵏᵗ

Slope field

Short segments showing dy/dx at many points.

Verifying a DE solution

Substitute the function and its derivatives and check both sides match.

Constant of integration (C)

Add it right after integrating, before solving for y.

Differential equation

An equation relating a function and its derivatives.

General solution

A family of solutions with a constant C.

Particular solution

The solution fitting a given initial condition.

Initial condition

A known value like y(0) = 3 used to find C.

Exponential growth/decay model

dy/dt = ky → y = y₀e^(kt).

Equilibrium solution

A constant solution where dy/dx = 0.

Sample practice questions

29 more questions like these, from level 1 to 5, are in the practice set.