MATH 246: Differential Equations

MATH 246 at the University of Maryland Global Campus is a three-credit introduction to the basic methods of solving differential equations.

UMGC lists first- and second-order equations with boundary-value problems, systems of linear differential equations, series solutions, existence and uniqueness, and formulating equations for physical systems.

MATH 141 (or MATH 132) is the prerequisite. Success depends on recognizing the type of equation and then applying the matching method accurately.

What MATH 246 Covers

Key Concepts Explained

Separable Equations

If dy/dx = xy, separate to dy/y = x dx, integrate to ln|y| = x^2/2 + C, and solve: y = A e^(x^2/2).

Second-Order Constant Coefficients

For y'' - 3y' + 2y = 0, the characteristic equation r^2 - 3r + 2 = 0 factors to (r - 1)(r - 2) = 0.

Modeling

If a quantity changes at a rate proportional to its size, dy/dt = ky, and the solution is y = C e^(kt). Setting up such an equation from words is often half the problem.

Typical Assignments and How to Approach Them

Classifying Before Solving

A simple first step is to name the equation. Is it first or second order? Linear or not? Separable? Constant coefficients? Once you can answer these, the method follows. Keep a table of types and methods, and add an example for each.

Always Check

You can verify any proposed solution by substituting it into the equation and the given conditions. Doing so catches algebra slips and sign errors that would otherwise cost marks.

A Worked First-Order Linear Example

Solve y' + 2y = 4. An integrating factor e^(2t) gives (e^(2t) y)' = 4 e^(2t), so e^(2t) y = 2 e^(2t) + C and y = 2 + C e^(-2t).

Matching Method to Equation

Existence and uniqueness theorems tell you when an initial value problem has exactly one solution, so it is worth noting the conditions when you quote them.

A Short Self-Test

Use these to check your command of the basics. Verify each solution by substitution.

• Solve dy/dx = 3y with y(0) = 2.
• Solve y' = x y^2 by separation of variables.
• Find the general solution of y'' - 4y = 0.
• Explain what existence and uniqueness mean for an initial value problem.
• Write a differential equation for a tank that loses liquid at a rate proportional to the volume.

Writing the verification line for each answer builds the habit that protects marks in longer problems.

Modeling Physical Systems

The catalog includes formulating equations for physical systems. Start from the law that governs the situation, such as a rate of change being proportional to an amount, write it as an equation, and name every symbol with units. Then solve, apply the initial condition and interpret the answer in the original context, noting any limits on where the model is reasonable.

Where Students Get Stuck

• Choosing the method. Classify first.
• Constants. Track constants of integration carefully.
• Applying conditions. Use them after finding the general solution.
• Interpretation. State what the solution means physically.

Study Tips for MATH 246

• Review integration techniques from MATH 141.
• Verify every solution by substitution.
• Make a table of equation types and methods.
• Practice modeling sentences into equations.

How We Help with MATH 246

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