DIFFERENTIAL EQUATIONS ACTUAL TEST
QUESTIONS AND SOLUTIONS
COMPREHENSIVE REVIEW PACKAGE
●● Order of a Differential Equation
Answer: The order of the highest derivative that occurs in the equation.
●● Solution to a Differential Equation
Answer: A function y = f (x) which, when substituted for y, makes the
equation make sense.
●● General Solution
Answer: A differential equation can have many solutions, for example y
= f (x) + c.
●● Verifying Solutions
Answer: To verify a solution, you can use the provided answer; to show
something, you must derive it yourself.
●● Initial Value Problems
Answer: Real-life problems usually have an initial value or initial
condition, leading to a unique solution like y = f (x) + 10.
, ●● Example of Verifying a Solution
Answer: Verify that the function y = 2/3ex + e−2x is a solution to the
differential equation y′ + 2y = 2ex.
●● Particular Solution
Answer: A unique solution that satisfies the initial condition, derived
from the general solution.
●● Exercise: Initial Value Problems
Answer: Verify that y = −t cos(t) − t + c is a general solution to the
differential equation sin(t)dy/dt - sin(t)dt + cos(t)d²y/dt² = t + sin(2t).
●● Initial Condition
Answer: A specific value that the solution must satisfy, such as y(π) = 3.
●● Verification Process
Answer: The process of substituting a function into a differential
equation to confirm it is a solution.
●● Example of Initial Value Problem
Answer: Find the particular solution given the initial condition y(π) = 3
for the general solution.
QUESTIONS AND SOLUTIONS
COMPREHENSIVE REVIEW PACKAGE
●● Order of a Differential Equation
Answer: The order of the highest derivative that occurs in the equation.
●● Solution to a Differential Equation
Answer: A function y = f (x) which, when substituted for y, makes the
equation make sense.
●● General Solution
Answer: A differential equation can have many solutions, for example y
= f (x) + c.
●● Verifying Solutions
Answer: To verify a solution, you can use the provided answer; to show
something, you must derive it yourself.
●● Initial Value Problems
Answer: Real-life problems usually have an initial value or initial
condition, leading to a unique solution like y = f (x) + 10.
, ●● Example of Verifying a Solution
Answer: Verify that the function y = 2/3ex + e−2x is a solution to the
differential equation y′ + 2y = 2ex.
●● Particular Solution
Answer: A unique solution that satisfies the initial condition, derived
from the general solution.
●● Exercise: Initial Value Problems
Answer: Verify that y = −t cos(t) − t + c is a general solution to the
differential equation sin(t)dy/dt - sin(t)dt + cos(t)d²y/dt² = t + sin(2t).
●● Initial Condition
Answer: A specific value that the solution must satisfy, such as y(π) = 3.
●● Verification Process
Answer: The process of substituting a function into a differential
equation to confirm it is a solution.
●● Example of Initial Value Problem
Answer: Find the particular solution given the initial condition y(π) = 3
for the general solution.
