Differential equations Questions and Answers

The function y 100e 0 045t models the amount y of radioactive material remaining in grams as a function of time t in days Calculate the half life of this radioactive material Round your answer to the nearest tenth 1 A B I NE

2 A company produces and sells electric cars The company s profit P in thousands of dollars changes based on the number of cars x they produce per month The rate of change of their profit from producing x electric cars is modelled by dP 1 6x 48 x 0 dx The company makes a profit of 260 thousand dollars when they produce 15 electric cars a Find an expression for P in terms of x P P 10 441 200 000 1605 43

A coolant is injected into engine fluid causing the temperature of the engine fluid to decrease with respect to time at a rate that is proportional to the difference of the fluid s instantaneous temperature T and the fluid s original temperature To Select the differential equation that represents the relationship 10 points OdT c T T dt dt c T T T t T To T t To T

Find the general solution of the given differential equation. x²y + xy = 4 Give the largest interval over which the general solution is defined. (Think about the implications of any singular points. Enter your answer using interval notation) Determine whether there are any transient terms in the general solution. (Enter the transient terms as a comma-separated list; if there are none, enter NONE.)

Find an explicit solution of the given initial-value problem. x2dy/dx = y-xy, y(-1) = -6

Consider the following differential equation and seek a power series solu- tion centered at ro = 0. (x²+4)y" - xy + 5y = 0 By putting the equation in standard form, we can see that the radius of convergence is at least (A) 2 (B) 4 (C) 5 (D) 6

Find the first four nonzero terms in a power series expansion about x=0 for the solution to the given initial value problem. w" +5xw'-w=0; w(0)=4, w'(0) = 0

Use implicit differentiation to find the tangent line of the function In(y) = x ln(x) at the point (2,4).

For the differential equation dy/dx =e-x-3x²: Part A: Find a general solution. Show all your work. Part B: Verify the solution. Show all your work.

If f(x) = 2² +1, use the limit definition of the derivative to compute f'(z). You can use derivative rules to check your answer, but answers calculated without using the definition of derivative will receive very little credit.

Solve the initial value problem x'(t) = Ax(1) for 120, with x(0)=(3,1) Classify the nature of the origin as an attractor, repeller, or saddle point of the dynamical system described by x¹=Ax. Find the directions of greatest attraction and/or repulsion 2 -4 6-8

5. Consider differential equation dy/dx + (2/x) y = xy³, y(1) = 1/2 Find y(10) numerically using the following methods and h = 0.5, 0.25, 0.125 and calculate the errors in each case. You have to use MATLAB for this problem. (10 Points each) a. Forward Euler's method b. Backward Euler's method c. Modified Euler's method d. Improved Euler's method e. Fourth-Order Runge Kutta Method

A jogger runs around a circular track of radius 65 ft. Let (x, y) be her coordinates, where the origin is the center of the track. When the jogger's coordinates are (39, 52), her x-coordinate is changing at a rate of 16 ft/s. Find dy/dt. dyldt = ft/s

The rate of change of the number of raccoons N(t) in a population is directly proportional to 380 N(t), where t is the time in years. When t = 0, the population is 120, and when t = 4, the population has increased to 160. Find the population when t = 8. (Round your answer to the nearest integer.) raccoons

Given the sets of system linear of differential equation; 2x"+6x'+y" + 3y' = 12e ^-t x'+2x + y' = 0 (a) Determine x(t) by using Elimination Method. (b) Determine y(t). (c) Check number of arbitrary constant and determine the final solutions.

In drawing the slope field for the differential equation dy/dx= x+3y-4, I would draw a short line segment for different points on the x-y plane. What slope should I draw the line segment at the point (0,-2)? Your Answer: