First-Order Equations: Separable, Linear and Exact: every key term you need (+ practice quiz)
25 flashcard terms for Differential Equations Topic 1, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
An equation relating an unknown function of one variable to one or more of its derivatives. Ordinary distinguishes it from a partial differential equation, where the unknown depends on several variables and partial derivatives appear.
Order of a differential equation
The order of the highest derivative that appears. A first-order equation involves y prime only; a second-order equation involves y double prime. Order sets how many arbitrary constants a general solution carries.
Linear differential equation
An equation in which the unknown function and its derivatives appear only to the first power and are never multiplied together or fed into another function. Coefficients may depend on the independent variable in any way at all.
Nonlinear differential equation
Any equation that fails linearity, for instance because it contains y squared, y times y prime, sine of y, or a square root of y. Nonlinear equations rarely admit closed-form solutions and often require qualitative or numerical study.
General solution
A family of solutions containing arbitrary constants, one constant for each order of the equation, that captures every solution obtainable by the method used. Fixing the constants from side conditions picks out one member of the family.
Particular solution
A single member of the solution family, obtained by assigning definite values to the arbitrary constants. In practice these values come from an initial condition or from a boundary requirement stated with the problem.
Initial value problem
A differential equation together with the value of the unknown, and of enough of its derivatives, at one chosen point of the independent variable. The side data are all imposed at the same point, which is what makes it initial rather than boundary.
Separable equation
A first-order equation that can be written as y prime equal to a function of x times a function of y. Dividing by the y factor and integrating each side against its own variable produces an implicit relation between x and y.
Separation of variables procedure
Move all y dependence with dy to one side and all x dependence with dx to the other, integrate both sides, add a single constant, then solve for y when possible. Division by a factor can hide solutions where that factor vanishes.
Lost equilibrium solution
A constant solution deleted during separation because you divided by an expression that is zero there. After separating, always check every root of the divided factor and reinstate it as a genuine solution of the equation.
First-order linear standard form
The arrangement y prime plus p of x times y equal to q of x, with the leading coefficient scaled to one. Every integrating-factor formula assumes this form, so rescaling before applying the method is not optional.
Integrating factor
The multiplier mu of x equal to the exponential of the integral of p of x. Multiplying the standard-form equation by it turns the left side into the exact derivative of mu times y, after which one integration finishes the problem.
Why the integrating factor works
It is engineered so that mu prime equals p times mu. That identity makes mu times y prime plus mu times p times y collapse into the product-rule derivative of mu times y, converting the equation into a direct antidifferentiation.
Constant of integration in the integrating factor
It may be dropped when forming mu, because an extra multiplicative constant cancels from both sides of the final formula. Carrying it costs effort and changes nothing about the resulting solution family.
Exact equation
A first-order equation written as M dx plus N dy equal to zero in which M dx plus N dy is the total differential of some potential function. Solutions are then the implicit level curves of that potential.
Exactness test
On a simply connected region, M dx plus N dy is exact exactly when the partial of M with respect to y equals the partial of N with respect to x. Checking this mixed-partial condition is the first step before hunting a potential.
Potential function reconstruction
Integrate M with respect to x, allowing an unknown function of y as the constant, differentiate the result with respect to y, and match it against N. That comparison determines the unknown function up to a numerical constant.
Integrating factor for inexactness
A multiplier chosen to make a nonexact equation exact. When the expression for the difference of the mixed partials divided by N depends on x alone, an integrating factor depending only on x exists and is found by a simple integral.
Bernoulli equation
A first-order equation of the form y prime plus p of x times y equal to q of x times y to the power n. The substitution v equal to y to the power one minus n converts it into a linear equation in v.
Homogeneous-degree substitution
When y prime depends on x and y only through the ratio y over x, setting v equal to y over x and writing y as v times x turns the equation into a separable equation for v as a function of x.
Implicit solution
A relation between x and y that defines the solution without expressing y as an explicit formula. Separable and exact methods often stop here, and forcing an explicit form can quietly select the wrong branch.
Direction field
A grid of short line segments whose slopes are given by the right-hand side of y prime equal to f of x and y. Solution curves must follow the segments, so the picture reveals qualitative behaviour with no formula in hand.
Isocline
The set of points where the direction field has a chosen fixed slope, obtained by setting f of x and y equal to a constant. Sketching several isoclines is a fast hand method for drawing an accurate direction field.
Autonomous first-order equation
An equation y prime equal to f of y whose right side does not mention the independent variable. Its direction field is constant along horizontal lines, and any solution shifted in the independent variable is again a solution.
Interval of validity
The largest open interval containing the initial point on which the solution exists and stays differentiable. It can be far shorter than the domain of the formula, since blow-up or division by zero truncates it.