- Recognize differential equations and distinguish their function-valued answers from answers to regular algebraic equations.
- Classify differential equations by derivative type, order, and linearity.
- Distinguish families, particular solutions, trivial solutions, explicit solutions, implicit solutions, singular solutions, and general solutions.
- Verify a proposed solution and identify an interval of definition.
- Use an initial condition to select a parameter value and an interval of definition.
Differential Equation Foundations
A regular algebraic equation can have a number as its answer. A differential equation has a function as its answer.
The functiony = c₁ cos(x) + c₂ sin(x)solves this differential equation.
What kind of answer does a differential equation have, in contrast with a regular algebraic equation?
A differential equation has a function as its answer, whereas a regular algebraic equation can have a number as its answer.
Types and Order of Differential Equations
| Derivative | Interpretation |
|---|---|
dy/dx | The rate of change of y with respect to x and the slope of the tangent line. |
∂u/∂x | The change in u in the x-direction. |
∂u/∂y | The change in u in the y-direction. |
- d²y/dx² + = sin(4x) is second order.
- (dy/dx)² + xy = eˣ is first order even though its first derivative is squared.
Classify the order of (dy/dx)² + xy = eˣ.
It is first order, even though the first derivative is squared.
Linear and Nonlinear Equations
In an equation that is linear in y, y and all its derivatives occur only to the first power. The coefficients multiplying y and its derivatives are functions only of x.
| Equation | Classification and reason |
|---|---|
y'' + 3xy' + x³y = 0 | Linear in y. |
y''' + 3yy'' + y = eˣ | Nonlinear because y multiplies y''. |
(dy/dx)² + xy = eˣ | Nonlinear because dy/dx is squared. |
x²y' + cos(x + y)y = 17 | Nonlinear because cos(x + y) is not a function only of x. |
xy'' + yy' = 0 | Nonlinear because y multiplies y'. |
Classify x²y' + cos(x + y)y = 17 as linear or nonlinear in y, and state the cited reason.
It is nonlinear because cos(x + y) is not a function only of x.
Classifications of Solutions
| Term | Meaning |
|---|---|
| One-parameter family | y = 3x² + C is a one-parameter family when each choice of C gives a solution. |
| Two-parameter family | y = c₁e²ˣ + c₂e⁴ˣ is a two-parameter family of solutions. |
| Particular solution | A solution obtained by choosing specific values for the arbitrary parameters in a family. |
| Trivial solution | The solution y = 0. |
| Term | Meaning |
|---|---|
| Explicit solution | y is solved explicitly in terms of x. |
| Implicit solution | y is implicitly defined in terms of x without being isolated. |
| Singular solution | A solution that cannot be obtained by choosing parameter values from the stated family. |
| General solution | A parameterized family from which every solution can be obtained by choosing parameter values. |
How does a singular solution differ from a general solution?
A singular solution cannot be obtained by selecting parameter values from the stated family. A general solution is a parameterized family from which every solution can be obtained.
Verifying Solutions and Finding Intervals
To verify a proposed solution, compute the required derivatives and substitute them into the differential equation.
The domain (-∞, 0) ∪ (0, ∞) is not an interval: it omits 0 between its negative and positive members. Without an additional condition selecting one side, either (-∞, 0) or (0, ∞) can be chosen as an interval of definition for y = 1/x.
With no additional condition, which intervals of definition can be chosen for y = 1/x?
Either (-∞, 0) or (0, ∞) can be chosen.
Initial Value Problems
An initial condition supplies information that can determine the arbitrary parameter in a one-parameter family. The condition = y₀ selects the solution curve passing through (x₀, y₀).
For the same family, = 1/3 also gives C = -1, while its selected interval of definition is (1, ∞).
For y = 1/(x² + C), what parameter and interval are selected by = 1/3?
The condition gives C = -1, and the interval of definition is (1, ∞).
Common misconceptions
Watch out: Squaring a first derivative makes the equation second order.
Instead: The equation (dy/dx)² + xy = eˣ is first order even though its first derivative is squared.
Watch out: The full domain (-∞, 0) ∪ (0, ∞) of y = 1/x is one interval.
Instead: That set is not an interval because it omits 0 between negative and positive members. Without an additional condition, either (-∞, 0) or (0, ∞) can be chosen as an interval of definition.
Watch out: Every solution must be obtainable by choosing parameters from any stated family of solutions.
Instead: A singular solution is a solution that cannot be obtained by choosing parameter values from the stated family.
Lesson recap
- A differential equation contains an unknown function and one or more derivatives, and its answer is a function.
- Its order is the order of its highest derivative.
- When an equation is linear in y, y and its derivatives occur only to the first power, and their coefficients are functions only of x.
- A particular solution selects parameter values from a family, while a singular solution cannot be obtained from the stated family.
- Verification requires the needed derivatives and substitution into the differential equation.
- An initial condition can determine the arbitrary parameter in a one-parameter family.