Directions: Write the letter of the correct answer and show your solution in the test booklet.

1. Find the particular solution of the DE x²𝑦𝑑𝑥 + 𝑥 2 𝑦 2 𝑑𝑦 = 0; when 𝑥 = 2, 𝑦 = 1.

a. 2𝑥³ + 3𝑦² = 19 b. 3𝑥³ + 2𝑦² = 9 c. 2𝑦³ + 3𝑥² = 19 d. 2𝑥 + 𝑦² = 5
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2. What type of DE is the given equation (𝑥𝑦 − 2𝑦 )𝑑𝑥 − (4𝑥𝑦 − 𝑥 2 ) 𝑑𝑦 = 0?
2
2
I. Variable Separable DE II. Homogeneous DE III. Exact DE
a. I, II and III b. I and II c. II and III d. I and III
𝜕2 f 𝜕𝑓
3. Consider the DE 𝑥 2 + = 𝑘, the DE is a
𝜕𝑥 𝜕𝑦
a. 𝑜𝑟𝑑𝑖𝑛𝑎𝑟𝑦 𝐷𝐸 b. 𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝐷𝐸 c. 𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑖𝑐 𝐷𝐸 d. 𝑐𝑢𝑏𝑖𝑐 𝐷𝐸
4. The standard form of differential equations is
a. 𝑦′ + 𝑃(𝑦)𝑥 = 𝑄(𝑦) b. 𝑦′ + 𝑃(𝑥)𝑦 = 0 c. 𝑑𝑦/𝑑𝑥 = 𝑔(𝑥)𝑓( 𝑦) d. 𝑀(𝑥, 𝑦)𝑑𝑥 + 𝑁(𝑥, 𝑦)𝑑𝑦 = 0
5. Find the particular solution of the differential equation (2xy - sin y+ 3x2)dx + ( x² - xcosy )dy = 0
a. 𝑥²𝑦 + 𝑐𝑜𝑠 𝑥 − 𝑠𝑖𝑛 𝑦 = 𝐶 b. 𝑐𝑜𝑠 𝑥 − 𝑠𝑖𝑛 𝑦 = 𝐶 c. 𝑠𝑖𝑛 𝑦 − 𝑥²𝑦𝑐𝑜𝑠 𝑥 = 0 d. 𝑥 2 𝑦 − 𝑥𝑠𝑖𝑛 𝑦 + 𝑥 3 = 𝐶
6. Are equations that relate one or more functions and their derivatives.
a. Differential Models b. Differential Equations c. Integrodifferential Equations d. Differential functions
7. How will you know if equation is homogeneous DE?
𝜕𝑀 𝜕𝑁
a. M(x, y) = N(x, y) c. =
𝜕𝑦 𝜕𝑥
𝜕𝑀 𝜕𝑁
b. = d. if both M(x,y)and N(x,y) have the same degree
𝜕𝑥 𝜕𝑦
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8. Is the differential equation 𝑥𝑦 𝑑𝑥 + ( 𝑥2
+ 3𝑦²) 𝑑𝑦 = 0, exact? Why?
2
a. Yes because checking yields x c. Yes because checking yields -1
b. Yes because checking yields 2x d. no
𝑑𝑦 2𝑦
9. If you solve = , and give the values for x and y, the final answer is called a
𝑑𝑥 𝑥
a. general solution b. particular solution c. general function d. particular number
10. Find the general solution of the DE (2𝑥 + 𝑦)𝑑𝑥 + (2𝑦 + 𝑥)𝑑𝑦 = 0.
a. 𝑥 2 + 2𝑥𝑦 + 𝑦² = 𝑐 b. 𝑥𝑦² = −𝑐𝑙𝑛(𝑥𝑦) c. 𝑥 2 + 𝑥𝑦 + 𝑦² = 𝑐 d. 𝑥² + 3(𝑥² + 2𝑦²) = 𝑐
11. Which of the following equations is a variable separable DE?
a. (𝑥 + 𝑥² 𝑦) 𝑑𝑦 = (2𝑥 + 𝑥𝑦²) 𝑑𝑥 c. (𝑥 2 + 𝑥𝑦) 𝑑𝑥 – 2𝑦 𝑑𝑦 = 0
b. 2𝑦 𝑑𝑥 = (𝑥² + 𝑦) 𝑑𝑦 d. 𝑦² 𝑑𝑥 + (2𝑥 – 3𝑥𝑦) 𝑑𝑦 = 0
12. These are equations involving one independent variables.
a. Partial Differential Equations b. Differential Function c. Differential Equations d. Ordinary Differential Equations
13. Eliminate the arbitrary constants of 𝑦 = 𝐶1 𝑒 2𝑥 + 𝐶2 𝑒 5𝑥
a. 𝑦 ′′ + 𝑦 ′ − 2𝑦 = 0 b. 𝑦 ′′ − 6𝑦 ′ + 5𝑦 = 0 c. 𝑦 ′′ − 𝑦 = 0 d. 𝑦 ′′ − 7𝑦 ′ + 10𝑦 = 0
14. Which of the following equations is Exact DE?
a. (𝑦 − 𝑥²𝑦) 𝑑𝑦 = 𝑦²𝑥 𝑑𝑥 b. 2𝑦 𝑑𝑥 = (𝑥² + 1) 𝑑𝑦 c. (𝑥 2 + 𝑦²) 𝑑𝑥 – 5𝑥𝑦 𝑑𝑦 = 0 d. 𝑦² 𝑑𝑥 + (𝑥𝑦² – 3𝑦) 𝑑𝑦 = 0
15. Which one is not one of the conditions of linear ordinary differential equations?
a. The dependent variable y and its derivatives occur to the first degree only
b. No products of y and/or any of its derivatives appear in the equation
c. No transcendental functions of y and/or its derivatives occur
d. The independent variable y and its derivatives occur to the first degree only
16. Find the differential equations of all straight lines passing through the point (0, 0).
a. 1 + 𝑦′ = 0 b. 𝑦 = (𝑥 − 1)𝑦′ + 1 c. 𝑦 = (𝑥 − 3)𝑦′ + 2 d. 𝑦𝑑𝑥 – 𝑥𝑑𝑦 = 0
17. In the differential field dv/dx, x is what kind of variable?
a. dependent variable b. arbitrary variable c. parameter variable d. independent variable
18. What is the differential equation of the family of parabolas having their vertices at the origin and their foci on the x-axis?
a. 2𝑥 𝑑𝑦 – 𝑦 𝑑𝑥 = 0 b. 𝑥 𝑑𝑦 + 𝑦 𝑑𝑥 = 0 c. 2𝑦 𝑑𝑥 – 𝑥 𝑑𝑦 = 0 d. 𝑑𝑦 / 𝑑𝑥 – 𝑥 = 0
𝑑2𝑦 𝑑𝑦
19. Is the given differential equation 2 + = 𝑒 𝑥 linear?
𝑑𝑥 𝑑𝑥
a. Yes if dy is trigonometric b. Yes if + y is present c. yes d. no
3 2
𝑑3𝑦 𝑑2𝑦
20. The order and degree of the DE ( 3 ) + ( 2 ) = 0 is
𝑑𝑥 𝑑𝑥
a. 2,2 b. 3,3 c. 2,3 d. 3,2