MAT 275 Test 1 Practice

Sep 30, 2026

MAT 275 Test 1 Practice covers differential equations in a problem-set format, emphasizing classification and solution behavior rather than full derivations for some items. Topics include separability and linearity checks, constructing first-order linear ODEs with desired long-term behavior, equilibrium analysis and stability for nonlinear systems, and interpreting slope/direction fields to match the correct differential equation. Additional practice includes solving constant-parameter ODEs, IVPs via separation and integrating factors, and applying models such as salt-in-tank mixing, population dynamics, and Newton’s law of cooling.

LanguageEnglish

CategoryExam

FormatPDF

Pages8

Reading Time20 min

AccessFree to download

Determine separability and linearity

Construct first-order linear ODEs

Equilibrium and long-term behavior

Slope field interpretation

Direction field sketching and isoclines

Modeling with population and equilibria

Second-order ODE properties and parameter solutions

Solving with constants in nonlinear forms

Implicit solutions evaluation

Family of solution curves

Separation of variables for IVPs

Salt mixing IVPs

Population extinction times

Newton’s law of cooling

Exponential growth ratios

Logistic-type growth maximum rate

Integrating factor IVPs

MAT 275 TEST 1 PRACTICE 1. Determine if each of the following equations is separable (Yes or No), and /or linear (Yes or No) . Record your answer in the following table. Do not attempt to solve the equations. | Equation | Separable | Linear | | --- | --- | --- | | 􀝕 ′ = 􀝐~~ ~~~~ ~~1 | | | | 􀝕 ′ = 􀝐~~ ~~~~ ~~1 | | | | 􀝕 ′ = cos (􀝐􀝕) | | | | 􀝕 ′ − 􀝐􀝕 = 􀝐 3 | | | 2. a) Write down a first order linear ODE whose solutions all approach 􀝕 = 1. b) Write down a first order linear ODE such that solutions other than 􀝕 = −3 all diverge from 􀝕 = −3 . 3. Consider the ODE 􀝕 ′ = 􀝕 2 (􀝕 − 1)(􀝕 + 2) a) Determine all the equilibrium (constant) solutions and classify them as stable, unstable or semi-stable. b) If 􀝕(0) = −1, what will be the behavior of the solution as 􀝐 → ∞? c) If 􀝕 (0) = 0, what will be the behavior of the solution as 􀝐 → ∞? 4. Which of the following differential equations best represents the slope field at the right? a) 􀝕 ′ = 􀝔 + 􀝕 b) 􀝕 ′ = 􀝔 − 􀝕 c) 􀝕 ′ = 􀝕 − 􀝔 d) 􀝕 ′ = 􀝔􀝕 e) 􀝕 ′ = 􀯫􀯬 5. Consider the direction field below. a) i. Sketch the graph of the solution which has initial value y(0) = 1.5. ii. Sketch the solution which has initial value y(0) = 0.5. b) Use your sketch to estimate the value of y(1) for the solutions in (a) i. and (a) ii. c) Some solutions ofthis differential equation grow when t is large and some do not. On the graph, sketch the curve representing the boundary between these two behaviors. Continue it in both directions until it leaves the direction field box. Label this curve “isocline”. d) What happens to the two solutions in part (a) as 􀝐 → ∞? e) Consider the solution with initial condition y(0) = 1. Estimate the value of y(50) . Explain your reasoning. f) The direction field above comes from one of the following differential equations. Tell which one it is and why. I. 􀝕 ′ = (1 − 􀝕)􀝕 II. 􀝕 ′ = 􀝕􀝁 −􀯧 III. 􀝕 ′ = 􀝕 − 􀝐 6. The population of Whooping Cranes has been found to obey the equation 􀯗􀯗􀯬􀯧 = 􀝃 (􀝕) (in appropriate units) . Here's a graph of the function g(y) for y between −1 and 10. a) Identify the equilibrium solutions and classify them as stable, unstable or semi-stable. b) For what values of the initial condition at t = 0 will the population be extinct? Give your answer in interval form. 7. Consider the differential equation: 􀝔2􀝕 ′′ − 3􀝔􀝕′ + 5􀝕 = 2􀝔2 ln (􀝔) (a) What is the order of the differential equation? (b) Is the differential equation linear or nonlinear? (c) Determine the value(s) of the constant A so that 􀝕(􀝔) = 􀜣􀝔2 ln (􀝔) is a solution to the differential equation. 8. Consider the differential equation: −4􀝔 + 􀝕 2 + 2􀝔􀝕􀝕′ = 0. (a) What is the order of the differential equation? (b) Is the Differential Equation linear or nonlinear? (c) Determine the value(s) of the constant 􀜣 so that 􀝕(􀝔) = 􀜣√􀝔 is a solution to the Differential Equation. 9. Determine the value(s) of the constant 􀝎 so that 􀝕 = 􀝁 􀯥􀯧 is a solution to 􀝕 ′′ − 􀝕 ′ − 6􀝕 = 0. 10. The general solution to a certain first order differential equation appears in implicit form 􀝕 2 − sin(􀝔) = 􀜥 . Determine the value at 􀝔 = 􀰗2 of the solution satisfying 􀝕(0) = −1 . 11. The general solution of the differential equation 􀝔􀝀􀝕 = 􀝕􀝀􀝔 is a family of (determine the correct choice) a) circles b) parabolas c) hyperbolas d) lines passing through the origin 12. Use separation of variables to find the solution of the following Initial Value Problems. Write your answer in explicit form and simplify as much as possible. For each IVP determine the interval in which the solution is defined. a) 􀝕 ′ = 􀯬21 , 􀝕 (0) = −2 b) 􀝔􀝕2 + 3􀝕2 − 􀝔 2􀝕 ′ = 0 , 􀝕 (1) = 3 c) 􀝕 ′ = 􀯧~~ ~~22􀯬􀯬 , 􀝕 (1) = 2 13. A tank initially contains 60 gal of pure water. Brine containing 1 lb of salt per gallon enters the tank at 2 gal/min, and the (perfectly mixed) solution leaves the tank at 3 gal/min; thus the tank is empty after exactly 1 hour. Let y(t) be the amount of salt in the t

FAQs

How do I decide whether an equation is separable or linear in this test practice?

For each given differential equation, evaluate whether it can be written in separated form for separability, and whether it is linear in the dependent variable and its derivatives for linearity. The practice explicitly asks for Yes/No entries without solving the equations.

What stability categories are used when analyzing equilibria in the nonlinear ODE problem?

Equilibrium solutions are classified as stable, unstable, or semi-stable. The test asks you to list the constant equilibrium solutions and determine their stability based on the ODE’s dynamics.

What methods are expected for solving the listed initial value problems?

The practice includes initial value problems designed for multiple methods: separation of variables for certain IVPs and the method of integrating factor for others. Some problems also require sketching solutions from direction fields rather than computing analytic forms.

Sep 30, 2026

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