First- and second-order differential equations model how quantities change in time, space, or another independent variable. This cheat sheet helps students recognize common equation types and choose a solution method quickly. It is especially useful for reviewing solution forms, initial conditions, and the behavior of linear systems.
Clear formulas reduce confusion when moving between algebra, calculus, and applications.
Key Facts
- A separable first-order equation has the form and is solved by rewriting it as before integrating.
- A linear first-order equation has the form and uses the integrating factor .
- After multiplying by the integrating factor, becomes .
- A homogeneous second-order linear equation with constant coefficients has the form and characteristic equation .
- If the characteristic roots are distinct real numbers and , then the general solution is .
- If the characteristic equation has a repeated real root , then the general solution is .
- If the characteristic roots are complex , then the general solution is .
- For a nonhomogeneous equation , the general solution is , where solves the homogeneous equation and is one particular solution.
Vocabulary
- Differential equation
- An equation involving an unknown function and one or more of its derivatives.
- Order
- The order of a differential equation is the highest derivative that appears, such as first order for or second order for .
- Initial condition
- A value such as or that is used to determine constants in a general solution.
- Integrating factor
- A function that turns a first-order linear equation into an exact derivative.
- Characteristic equation
- The algebraic equation obtained from a constant-coefficient homogeneous equation .
- Particular solution
- A single solution that satisfies a nonhomogeneous differential equation such as .
Common Mistakes to Avoid
- Forgetting the constant of integration in a first-order solution is wrong because the general solution needs an arbitrary constant such as before applying an initial condition.
- Using an integrating factor on an equation that is not in the form is wrong because must be the coefficient of after the coefficient of is .
- Solving incorrectly is serious because every second-order homogeneous solution depends on the correct characteristic roots.
- Using for a repeated root is wrong because the two terms are not independent, so the correct form is .
- Choosing a particular solution that duplicates part of is wrong because it will not be linearly independent, so the trial form must be multiplied by when overlap occurs.
Practice Questions
- 1 Solve the separable equation with initial condition .
- 2 Solve the first-order linear equation using the integrating factor method.
- 3 Find the general solution of .
- 4 Explain how the solution behavior changes when the characteristic roots of are real distinct, repeated, or complex.
Understanding Differential Equations First- and Second-Order
A differential equation does not usually give a number as its answer. It gives a rule for a whole family of possible functions. The constants that appear after integration represent the different starting situations that can follow the same rule.
Initial values select one member of that family. In a first-order problem, one initial value is normally needed. In a second-order problem, two pieces of starting information are normally needed, often a position and an initial rate of change.
This matches physical motion. Knowing where an object is at one instant is not enough to predict its later motion. Its velocity matters too.
Separation works only when every factor involving the dependent variable can be placed with the differential for that variable, while the remaining terms stay with the independent variable. Students often divide too quickly and accidentally lose a solution. If a factor involving the dependent variable can equal zero, test that case separately before dividing by it.
Integration can introduce logarithms, so absolute values matter when the variable may be negative. The final answer may be implicit rather than solved for the dependent variable.
That is still a valid solution if it correctly describes the relationship. Differentiating the final result is a reliable check because it should rebuild the original rate rule.
An integrating factor is built from the product rule. Its purpose is to turn the left side of a linear equation into the derivative of one product. This is why the method is more than a formula to memorize.
The coefficient of the unknown function controls the factor used to reshape the equation. After integration, divide by the integrating factor to recover the unknown function. The constant of integration must remain until an initial condition is used.
These equations appear in cooling, drug concentration, charging circuits, and population models with a steady input or loss. The equation separates the natural tendency of the system from an outside source that pushes it.
For second-order equations, the characteristic roots describe the basic motion patterns. A positive real root creates growth, while a negative real root creates decay. Two different roots give two independent growth or decay patterns.
A repeated root needs an extra factor involving the independent variable because one exponential alone cannot provide two independent solutions. Complex roots produce oscillation. Their real part controls whether the oscillations fade, stay at a fixed size, or grow.
This is the mathematics behind springs, vibrations in structures, and electrical signals. When a forcing term is present, the homogeneous part describes the system's own behavior and the particular part describes its response to the input.
Pay close attention when a trial particular solution duplicates a homogeneous term. In that case, multiply the trial by a suitable power of the independent variable to make it independent.