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# Discuss properties of the solution of a differential equation

Consider the differential equation

Let be a solution to the equation with the initial condition . Without attempting to solve this equation explicitly answer the following questions.

1. Since we also have . Does have a relative maximum, relative minimum, or neither at 0?
2. If then and if then . Find positive numbers and such that

3. Prove that

4. Prove that

for some finite number and find the value of .

Incomplete.

# Find the launch velocity necessary for a rocket to escape Earth only considering gravity

Ignoring all forces except the Earth’s gravity find the speed with which you must launch a rocket so that it never returns to Earth.

Incomplete.

# Find and solve a differential equation governing population growth with given conditions

The population of a town at time is 365. The population growth factor is , and the town experiences a death rate of one percent of the population per day. Find a differential equation modeling the population of the town as a function of time and find

1. the actual population of the town after years,
2. the cumulative total of the fatalities from the town’s death rate.

Incomplete.

# Find and solve a differential equation for the decay of a material with given properties

A given substance decays at a rate proportional to the squareroot of the amount of the material present. Show that a substance with this decay property will completely decay in a finite amount of time and find this time.

Incomplete.

# Find and solve a differential equation for the decay of a material with given properties

A given substance decays at a rate proportional to the square of the amount of the material present. At the end of one year there is 0.5 grams of the substance remaining.

1. Create and solve a differential equation that governs the mass of the material present after years.
2. Find the decay constant of the material in units .

Incomplete.

# Prove a substitution converts a given second order equation to a first order equation

1. Consider the second-order differential equation

and let be a solution to the equation. Show that the substitution converts the equation

into a first-order liner equation for .

2. By inspection, find a nonzero solution of the second order differential equation

and use part (a) to find a solution of the differential equation

with

Incomplete.

# Find solutions of a differential equation of a given form

1. Let be a function such that

Let and show that satisfies

for constants . Determine the values of the constants.

2. Find a solution .

Incomplete.

# Solve the differential equation (1 + y2e2x) y′ + y = 0

Consider the differential equation

Make a change of variable where is a function of and is a constant and solve the differential equation.

Incomplete.

# Solve the differential equation (x + y3) + 6xy2y′ = 0

Use a change of variables to convert the following differential equation into a linear differential equation, and then solve the equation:

Incomplete.

# Find the solution of the differential equation xy′′ – y′ + (1 – x)y = 0

Assume the differential equation

has a solution of the form

for some constant . Determine an explicit formula for this solution.

Incomplete.