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Test 1

This document is a test paper for the MTC22ES course at the Central University of Technology, Free State, covering differential equations and their classifications. It includes instructions for answering questions, with a total of 60 marks available across two main questions. The test assesses knowledge on various types of differential equations, including variable separable, exact, linear, homogeneous, and Bernoulli equations.

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22 views3 pages

Test 1

This document is a test paper for the MTC22ES course at the Central University of Technology, Free State, covering differential equations and their classifications. It includes instructions for answering questions, with a total of 60 marks available across two main questions. The test assesses knowledge on various types of differential equations, including variable separable, exact, linear, homogeneous, and Bernoulli equations.

Uploaded by

93nhlanhlambanjwa
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Central University of Technology, Free State Department of Mathematics, Science and Technology Education Welkom Campus MTC22ES Test 1: Semester 2 2024 Duration: 80 minutes Total : 60 Marks Instructions 1. Answer ALL questions. | 2. Show all working. 3. Write clearly and keep all parts of the solution together. 4. Number your answers correctly. 5. Marks are shown to the right of each question. Question 1: [46 Marks] Classify each of the following differential equations as either _ variable separable, exact, linear, homogeneous or Bernoulli. (a) ydy | rt 9 ane < a dy _ ty = ==4$it1 dx yo a nytt ty =e Q variable separable equations (aA o=* A © (a) a a--! o ly da®y? + y (1) = de = ype age YO) (b) a = Py, yQ) =0 {6 (1.3) Solve the following exact differential equations (a) (€" — ycosary)dx + (2ne% — x cos zy + 2y)dy = 0 16) (b) (2c — 1)(y — 1)de + (x + 2)(x — 3)dy =0, y(1) = —-1 (5) Determine the value of k which makes the following differential equa- tion exact ; (ye™” + x)de + kwe*Ydy = 0 - (2) the following inexact differential equation (82?y + 2xy + y*)de + (a? + y2)dy = 0 5] homogeneous differential equation dy y 1 Parlay paelad i gtgvt ty 4 Tinear differential equations d (42) tate +e =o Mm Ao 1.7) Solve the following Bernoulli equation (2y° — adr + 2ey2dy = 0, x #0, y(l)=1 Question 2: [14 Marks] of hot water with an initial temperature of 80°C, is placed in where the temperature is - After one minute, the water drops to 70°C. will be the temperature of the water after 3 minutes? (6) Jong does it take for the water to cool down to 40°C? (2) culture of bacteria grows at arate in 4 days, find the time requi its original size. Proportional to its size. If the red for the culture to increase (6 END OF TEST

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