Questions: Introduction to Differential Equations

Question 1

What is the general solution to the differential equation dy/dx = y?

• A. y = x + C
• B. y = Ce^x
• C. y = x²/2 + C
• D. y = ln(x) + C

Answer: (B)

We need a function whose derivative equals itself. The exponential function y = e^x has this property. Scaling by any constant C still satisfies the equation: d(Ce^x)/dx = Ce^x = y. The general solution is y = Ce^x, where C is determined by an initial condition. The other options do not satisfy dy/dx = y.

Question 2

The solution to a differential equation is generally a single specific number.

• T. True
• F. False

Answer: False

Solutions to differential equations are functions, not numbers. The general solution is typically a family of functions parameterized by one or more constants (e.g., y = Ce^x). A particular solution is one member of that family, selected by applying an initial or boundary condition. The constants arise because integration — the reverse of differentiation — always introduces an arbitrary constant.

Question 3

What is the difference between a general solution and a particular solution of a differential equation?

Model answer: A general solution represents the full family of functions satisfying the equation, containing arbitrary constants. A particular solution fixes those constants using initial or boundary conditions to give one specific function.

When you integrate to solve a differential equation, each integration step introduces one arbitrary constant. The general solution preserves these constants. To find a particular solution, you apply given conditions (e.g., y(0) = 3) to determine the constant values, selecting the single function from the family that satisfies both the equation and the conditions.