Math Problem Statement

Solve the problem

Find the general solution x(t) of the differential equation m(d^2x/dt^2) + r(dx/dt) - mg = 0, where m is the mass of the ball, r is a constant related to the viscosity of the oil, g is the magnitude of the acceleration due to gravity, and x is the vertical distance from the point of release.

Solution

The differential equation provided is:

md2xdt2+rdxdtmg=0m \frac{d^2 x}{dt^2} + r \frac{dx}{dt} - mg = 0

This is a second-order linear ordinary differential equation with constant coefficients. The hint indicates that the auxiliary polynomial has two real roots, suggesting a particular solution structure.

Step 1: Homogeneous Form

To solve this, we begin by rewriting the equation in standard form:

d2xdt2+rmdxdtg=0\frac{d^2 x}{dt^2} + \frac{r}{m} \frac{dx}{dt} - g = 0

Now, the equation is:

d2xdt2+rmdxdt=g\frac{d^2 x}{dt^2} + \frac{r}{m} \frac{dx}{dt} = g

Step 2: Solving the Homogeneous Equation

First, let's solve the homogeneous part of the equation, i.e., when g=0g = 0:

d2xdt2+rmdxdt=0\frac{d^2 x}{dt^2} + \frac{r}{m} \frac{dx}{dt} = 0

The auxiliary equation for this differential equation is:

m2+rmm=0m^2 + \frac{r}{m}m = 0

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Math Problem Analysis

Mathematical Concepts

Differential Equations
Second-Order Linear ODEs

Formulas

m(d^2x/dt^2) + r(dx/dt) - mg = 0
Auxiliary equation: mλ^2 + rλ = 0

Theorems

Solution of second-order differential equations with constant coefficients

Suitable Grade Level

Undergraduate level (Calculus/Differential Equations)