Differential Equations quiz

The order of the differential equation y'' + 5y' + 6y = 0 is:

The degree of (y'')^3 + (y')^2 + y = 0 is:

The integrating factor of dy/dx + y = e^x is:

The number of arbitrary constants in the general solution of a third order differential equation is:

Which function is a solution of y' = y?

The integrating factor of dy/dx - y/x = x^2 for x > 0 is:

The general solution of dy/dx = 2x is:

A particular solution of a differential equation contains:

The general solution of dy/dx = e^(x - y) is:

The substitution used to solve a homogeneous differential equation dy/dx = F(y/x) is:

Find the general solution of dy/dx = sec^2 x. (2 marks)

Solve dy/dx = y/x for x, y > 0. (2 marks)

Find the integrating factor of x dy/dx - y = x^4 - 3x for x > 0. (2 marks)

Solve dy/dx = (1 + y^2)/(1 + x^2). (2 marks)

Verify that y = sin x is a solution of y'' + y = 0. (2 marks)

Find the order and degree of (dy/dx)^2 + 3y = cos x. (2 marks)

Solve dy/dx + 3y = 0 given y = 2 when x = 0. (2 marks)

Find the general solution of y log y dx - x dy = 0. (2 marks)

Is the equation (x^2 + y^2) dx - 2xy dy = 0 homogeneous? Justify. (2 marks)

Find the general solution of dy/dx = e^(2x) + x. (2 marks)

Find the general solution of dy/dx = e^(x + y). (3 marks)

Find the particular solution of dy/dx = 2xy given that y = 3 when x = 0. (3 marks)

Solve dy/dx + y = e^(-x) given that y = 1 when x = 0. (3 marks)

The rate of growth of a bacteria population is proportional to the number present. The population increases by 10% in 2 hours. In how many hours will it double (log 2 = 0.6931, log 1.1 = 0.0953)? (3 marks)

Show that y = Ae^(2x) + Be^(-x) satisfies y'' - y' - 2y = 0. Find A and B if y = 3 and y' = 0 when x = 0. (3 marks)

Read the passage and answer the questions. A cup of tea at 85 degree C is placed in a room at 25 degree C. By Newton's law of cooling, dT/dt = -k(T - 25). After 5 minutes the tea is at 65 degree C. (i) Solve the equation to get T in terms of t. (ii) Find e^(-5k). (iii) Find the temperature after 10 minutes. (5 marks)

Read the passage and answer the questions. A curve passes through the point (1, 3) and its slope at any point (x, y) is 2y/x. (i) Write the differential equation. (ii) Find its general solution. (iii) Find the equation of the curve. (5 marks)

Read the passage and answer the questions. A small stone falls through a fluid. Its velocity v (m/s) satisfies dv/dt = 10 - 2v, with v = 0 at t = 0. (i) Identify the type of differential equation. (ii) Solve it to find v in terms of t. (iii) Find the limiting (terminal) velocity as t becomes very large. (5 marks)

Read the passage and answer the questions. A radioactive substance decays at a rate proportional to the amount present, dN/dt = -kN. Its half-life is 1,600 years. (i) Solve the differential equation. (ii) Find k in terms of log 2. (iii) What fraction remains after 3,200 years? (5 marks)

Read the passage and answer the questions. A teacher writes three equations on the board: (P) dy/dx = x^2 y^2, (Q) dy/dx = (x^2 + y^2)/(2xy), (R) dy/dx + y/x = x. (i) Classify each equation by type. (ii) State the method for each. (iii) Find the integrating factor of (R). (5 marks)

Find the particular solution of (1 + e^(2x)) dy + (1 + y^2) e^x dx = 0, given that y = 1 when x = 0. (6 marks)

Solve the homogeneous differential equation x dy - y dx = sqrt(x^2 + y^2) dx, for x > 0. (6 marks)

Find the particular solution of dy/dx + y cot x = 2x + x^2 cot x, given that y = 0 when x = pi/2. (6 marks)

Find the particular solution of dy/dx - 3y cot x = sin 2x, given that y = 2 when x = pi/2. (6 marks)

Find the equation of the curve passing through (0, -2) such that the product of the slope at any point and its y coordinate equals its x coordinate. Sketch the curve. (6 marks)