Differential Equations Answers

Solve the following differential equation: 2dy/dx + y = y^3(x - 1)

Solve the following differential equations: (x^3+y^3)=(xy^2)dy/dx

Activity 1:

Solve the following problems applying the concepts learned above. Write your answer on a separate sheet of paper. (Show your solution)

1. The rate of change x is proportional to x. When t =0, x0 = 3 and when t =2, x=6. What is the value of x when t =4?

2. A certain plutonium isotope decays at a rate proportional to the amount present. Approximately 15% of the original amount decomposes in 100 years. How much amount of the substance has decayed after 600 years? Also, find the half - life t1/2 of this radioactive substance ; that is, find the time required for this substance to decay to one- half of its original amount.

integrating factor by inspection (2x^2 + y)dx + (x^2y - x)dy = 0

Show that simple harmonic motion y(t) = C1 cos ωt + C2 sin ωt can be written as:

a. y(t) = A sin(ωt + ϕ0)

b. y(t) = A cos(ωt + ϕ1)

Verify that 𝑦1(𝑥) = 1 and 𝑦2(𝑥) = 𝑥^(1/2) are solutions of the differential equation 𝑦𝑦′′ + (𝑦′)^2 = 0 for 𝑥 > 0. Then show that 𝑦 = 𝑐1 + 𝑐2𝑥^(1/2) is not, in general, a solution to the equation. Explain why this does not contradict superposition principle.

Solve the following using the method of undetermined coefficients. (d ^ 2 * y)/(d * t ^ 2) + (dy)/(dt) - 5y = - 6x ^ 3 + 3x ^ 2 + 6x

A 100-volt electromotive force is applied to an RC series circuit in which the resistance is 200 ohms and the capacitance is 10−4 farad. Find the charge q(t) on the capacitor if q(0) = 0. Find the current i(t).

solve system of linear differential equations  dx/dt-4y=1 and dy/dt+x=2