Scanned By Cam Scan Type Text Assignment I

Question

Scanned By Camscannertype Texttype Texttype Textassignment Ins Scanned by CamScanner [Type text] [Type text] [Type text] 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

Introduction Differential equations are fundamental tools in modeling various physical systems, including mechanical vibrations and fluid dynamics. This paper addresses four distinct problems: solving differential equations, verifying solutions, modeling spring vibrations with damping, and modeling salt concentration dynamics in a tank. Each problem requires the formulation, solution, and interpretation of results within real-world contexts. Problem 1: Solving a Differential Equation and Verification The first task involves solving a given differential equation. Although the specific equation is not explicitly presented in the instructions, a typical example might be a first or second-order linear differential equation such as $\frac{dy}{dx} + p(x)y = q(x)$ or a homogeneous second-order equation like $m y'' + c y' + k y = 0$. The general approach includes separating variables where applicable, integrating, and finding the particular solution that satisfies boundary conditions. Suppose, for illustration, the differential equation is $\frac{dy}{dx} = y$. The solution can be obtained via separation of variables: $$ \frac{dy}{y} = dx \implies \ln|y| = x + C \implies y = Ae^{x}, $$ where $A = e^C$ is determined by initial conditions. Verification involves substituting the solution back into the original differential equation to confirm that both sides are equal. Similarly, for more complex equations such as second-order linear ODEs, characteristic equations and auxiliary solutions are employed. Once the general solution is found, it should be verified by substitution. Problem 2: Solving an Initial Value Problem (IVP) and Verification Initial value problems specify conditions at a particular point, usually $x = x_0$, such as $y(x_0) = y_0$. Solving requires integrating differential equations, applying initial conditions to determine constants. For example, solving $\frac{dy}{dx} = 3x^2$ with $y(1) = 4$: $$ y = x^3 + C, $$ and using the initial condition: \

Scanned By Cam Scan Type Text Assignment I

Scanned By Camscannertype Texttype Texttype Textassignment Ins

Paper For Above instruction

Introduction

Problem 1: Solving a Differential Equation and Verification

Problem 2: Solving an Initial Value Problem (IVP) and Verification

Problem 3: Newton’s Second Law Model for a Damped Vibrating Spring

Problem 4: Salt Concentration in a Tank

Conclusion

References