Differential Equations

A differential equation relates an unknown function to one or more derivatives. A proposed solution is checked by substitution, not by its appearance. An initial condition selects a particular member of a solution family when the relevant existence and uniqueness conditions hold.

An equation such as y′ = ky states that the rate of change is proportional to the current quantity. A solution is a function satisfying the equation over an interval, not just a number making one algebraic statement true. A differential equation can have infinitely many solutions.

An initial condition selects a particular solution when appropriate existence and uniqueness conditions hold. Verify a proposed solution by differentiating and substituting into the equation, then separately checking the initial condition. A formula satisfying one but not the other is not the requested solution.

A slope field places short segments with slope determined by the differential equation at each plotted point. A solution curve follows those directions. For an autonomous equation y′ = f(y), slopes are the same along horizontal lines because they depend on y rather than directly on the independent variable.

Horizontal solution levels occur where the rate is zero. Sign analysis above and below such equilibria indicates whether nearby solutions move toward or away from them. A field gives qualitative behavior but is not a license to cross or join arbitrary solution curves.

For equations that can be written dy/dx = g(x)h(y), separate y-dependent factors and x-dependent factors, then integrate both sides. Include a constant and apply the initial condition. Algebraic rearrangement may divide by h(y), so check any solutions lost when h(y) = 0.

Absolute values in logarithmic antiderivatives matter. Use the initial condition and domain to choose a consistent branch when solving for y. A solution may have a finite interval of validity because it becomes undefined or violates the original model outside that interval.

The equation y′ = ky has solutions Ce^kx. Positive k gives growth for positive y, while negative k gives decay. Doubling time and half-life follow by comparing ratios, not by subtracting fixed amounts. The constant k has inverse-input units.

A real population, cooling system, or concentration may only approximate this behavior over a limited interval. Fit or derive k from the stated condition and keep units consistent. An equation proportional to y minus an ambient level produces exponential approach toward that level rather than toward zero.

The sign of y′ indicates increase or decrease, and differentiating the differential equation can reveal concavity along a solution. This may answer a question without an explicit formula. A slope field or rate equation can also show when a linear approximation should over- or underestimate locally.

State the physical domain, such as nonnegative time and positive population. Mathematical continuation into negative values or past a singularity may not represent the original situation. A model’s solution is only as meaningful as its assumptions.

Euler’s method follows the local tangent slope for a finite step: y(next) = y(current) + h f(x(current),y(current)). Update both x and y at every step. The method approximates a solution and usually depends on step size.

A smaller step often improves accuracy over a fixed interval, but the differential equation and numerical stability matter. For y′ = y, y(0)=1, two steps of h=0.5 give 1.5 and then 2.25, rather than the exact e at x=1.

The logistic model y′ = ky(1 − yL) for k > 0 and L > 0 has equilibrium solutions 0 and L. For 0 < y < L, the quantity increases; for y > L, it decreases toward L. The parameter L is a carrying level under the model.

For a growing solution between 0 and L, the growth rate is greatest at y = L2. This is a maximum of total growth, not per-capita growth. The solution is concave up below L2 and concave down above it, as follows by differentiating the rate relation.

Each segment in a slope field represents the derivative at a point. A solution curve must be tangent to the local segment as it passes through that point. Follow slopes from the initial condition rather than joining all horizontal segments together. Different initial conditions can produce different solution curves through the same field.

For y′=f(y), horizontal lines where f(y)=0 are equilibrium solutions. Test the sign of f(y) above and below each equilibrium to see whether nearby solutions move toward or away from it as x increases. A slope field does not show speed in the geometric sense of a particle unless the variables actually describe motion. It shows a rate of one variable with respect to another.

For a separable equation, rearrange so one side contains the y-expression with dy and the other contains the x-expression with dx. Integrate both sides and include a constant. Apply the initial condition before or after solving explicitly, as convenient, and verify the result in the original equation.

Dividing by a function of y can discard constant solutions where that function is zero. Check those equilibria separately. If integrating 1y gives ln|y|, the absolute value carries sign information on an interval avoiding zero. A logarithmic equation can be solved for an exponential expression with a signed multiplicative constant; the initial condition determines the appropriate sign and value.

The equation y′=ky says that the instantaneous rate is proportional to the current amount. Its solutions have the form Ce^kx. Positive k gives growth and negative k gives decay for positive initial amounts. The parameter k has inverse-time units; it is not automatically the percentage change over one whole time unit. The one-unit multiplicative factor is e^k.

To determine k from two measurements, divide the amounts to eliminate C, take a natural logarithm, and divide by elapsed time. Use the resulting model only over an interval where its assumptions are reasonable. Exponential growth does not include resource limits. A temperature-difference model can apply proportional decay to the difference from an ambient temperature rather than to the object’s absolute temperature.

An equation y′=F(x,y) gives a slope at each allowed point, not a single curve until an initial condition is supplied. A slope field shows short tangent segments. A solution curve should follow those slopes; it need not pass through every displayed point.

Verify a proposed solution by differentiating it and substituting both y and y′ into the equation. Then check the initial condition separately. A function can satisfy the differential equation but fail the required starting value.

Equilibrium solutions have constant y and therefore zero derivative. Find values that make the right side zero for all relevant x. Dividing by an expression involving y during separation may discard these solutions, so check them before dividing.

For a separable equation, rearrange to put y-dependent factors with dy and x-dependent factors with dx. Integrate both sides, retaining a constant. Apply the initial condition to identify the particular solution, then consider the interval where that solution is valid.

Logarithms often appear when integrating 1y. The absolute value and initial condition determine the appropriate branch when solving for y. A denominator becoming zero can limit the solution interval even when the original algebra seems to produce a formula for many x-values.

Implicit answers can be valid when solving explicitly is inconvenient, but they still need the initial condition and appropriate interpretation. Check by differentiation rather than trusting a sequence of rearrangements.

In y′=ky, the rate is proportional to the current amount. Positive k gives growth for a positive amount; negative k gives decay. The solution y=Ce^kx follows from separation, and C is determined by the initial value when x=0.

A model y′=k(M−y) approaches a limiting value M for positive k. The changing quantity is the distance from M. A negative derivative can become less negative as the solution approaches equilibrium. Do not infer constant rate merely because k is constant.

Interpret the model’s assumptions: proportionality, fixed parameters, and an appropriate domain. A mathematically exact solution may be a simplified description of a physical system. The model’s units require k to have inverse-input units when it multiplies y to produce y′.

For an autonomous equation y′=F(y), all points at the same height have the same slope. Horizontal rows of matching slope segments reveal this independence from x. Equilibrium heights occur where F(y)=0. Sign analysis above and below them predicts whether nearby solutions move toward or away from equilibrium.

If y′=2−y, slopes are positive below y=2 and negative above it. Solutions approach the equilibrium from either side in the standard model. The slope approaches zero as the solution approaches the limiting height. This graphical reasoning predicts behavior before separation produces a formula.

For a nonautonomous equation, slopes can vary with both x and y. Do not impose horizontal-row patterns when x appears explicitly. A solution passing through one point follows the field locally; drawing an arbitrary smooth curve through the initial point is insufficient.

An antiderivative constant is not optional decoration. It selects the solution that matches the initial condition. Substitute the initial coordinates into the integrated relation, solve for the constant, and then check the completed expression in the differential equation.

For y′=3y² with y(0)=1, separation gives −1y=3x+C, so C=−1 and y=1(1−3x). The formula has a singularity at x=13. The solution interval containing the initial point cannot simply pass through that singularity, even though the expression is defined again on the other side.

This illustrates why a solution includes a domain. Check denominators, logarithm arguments, and any divisions made while separating. A constant equilibrium solution may have been excluded by dividing by y or another factor and should be considered separately when appropriate.

Euler’s method repeatedly uses a tangent slope: y next=y current+hF(x current,y current). Update both coordinates after each step. Reusing the initial slope turns the process into one long tangent-line estimate rather than Euler’s method. A smaller step often improves approximation but does not make the method exact.

For y′=ky(1−yL) with positive k and L, the equilibria are 0 and L. Positive populations below L increase; those above L decrease. The greatest absolute growth occurs at y=L2 because the quadratic rate function is largest there. Per-capita growth is instead k(1−yL).

The logistic model assumes its parameters and limiting relationship remain appropriate. It is not a claim that real populations never overshoot or that carrying capacity cannot change. Use sign analysis and the initial value before choosing a solution sketch.