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Type Texttype Texttype Textassignment Instructionsfor Each Pr Assignment Instructions: For each problem, be sure to show all steps for arriving at your solution. Work within this document. Use an equation tool as needed, and submit everything in one file. 1. Solve the following differential equation, showing all work. Verify the solution you obtain. 2. Solve the following initial value problem, showing all work. Verify the solution you obtain. 3. Using the Newton’s second law model for a vibrating spring with damping and no forcing, , find the equation of motion if kg, kg/sec, kg/sec2, , and m/sec. What is the position of the mass after 1 second? Show all work. 4. A brine solution of salt flows at a constant rate of 6 L/min into a tank that initially held 50 L of brine solution into which was dissolved 1.5 kg of salt. The solution in the tank is stirred and flows out of the tank at the same constant rate of 6 L/min. If the concentration of the salt entering the tank is 0.05 kg/L, develop the differential equation that models this scenario and find an expression for the mass of salt, S ( t ), in the tank at time t minutes. Use that function to find the amount of salt in the tank after 10 minutes. Show all work.

Dr. Jack HW Helper · Accepted Answer

Solve the differential equation and verify the solution Suppose the differential equation is $\frac{dy}{dt} + py = q(t)$. To solve, we first find the integrating factor $ \mu(t) = e^{\int p dt} $. Multiplying through by this integrating factor, the equation becomes exact, and we can integrate both sides to find the solution. Assuming the specific differential equation is $\frac{dy}{dt} + 3y = 6t$, the integrating factor is $ \mu(t) = e^{3t} $. Multiplying through, $ e^{3t} \frac{dy}{dt} + 3 e^{3t} y = 6 t e^{3t} $. Recognizing the left side as the derivative of $ y e^{3t} $, we integrate both sides: $ y e^{3t} = \int 6 t e^{3t} dt + C $. Applying integration by parts to $ 6 t e^{3t} $, we obtain the integral and then solve for $ y(t) $. Verification involves differentiating $ y(t) $ and confirming it satisfies the differential equation. Solve the initial value problem Given an initial condition, say $ y(0) = y_0 $, substitute $ t=0 $ and $ y=y_0 $ into the solution to find the constant $ C $. For example, if $ y(0) = 2 $, then substituting back into the integrated form yields $ C $. Modeling and solving the mass-spring damping problem Newton's second law gives the differential equation $ m x'' + c x' + k x = 0 $, where $ m $ is mass, $ c $ is damping coefficient, and $ k $ is spring constant. For specified values, plug in the numbers: Suppose $ m = 1 \text{ kg} $, $ c=0.5 \text{ kg/sec} $, $ k=9.8 \text{ N/m} $. The equation becomes $ x'' + 0.5 x' + 9.8 x=0 $. To find position after 1 second, solve the characteristic equation, find the general solution, and evaluate at $ t=1 $ with initial displacement and velocity as conditions. Salt concentration problem with inflow and outflow Let $ S(t) $ be the salt amount at time t. The rate in is $ 0.05 \times 6 $ kg/min, the rate out is $ \frac{S(t)}{V(t)} \times 6 $ kg/min, where $ V(t) = 50 $ L (initial volume, constant). The differential equation is: \( \frac{dS}{d

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Sample Paper For Above instruction

Solve the differential equation and verify the solution

Solve the initial value problem

Modeling and solving the mass-spring damping problem

Salt concentration problem with inflow and outflow

References