Separable Differential Equations

Order of an equation and what counts as a solution on an interval, then splitting dy/dx = M(x)N(y) into g(y) dy = f(x) dx and integrating each side.

The ODE Project, Separable Differential Equations

Status

FlagsLoading your progress…

23 exercises

Exercise 2

Find the general solution for each equation in Finding General Solutions.

$x \, dx - y^2 \, dy = 0$

2.5

Exercise 3

Find the general solution for each equation in Finding General Solutions.

$\dfrac{dy}{dx} = \dfrac{y}{x}$

2.5

Exercise 4

Find the general solution for each equation in Finding General Solutions.

$(t^2 + 1) \, dt + (x^2 + x) \, dx = 0$

2.5

Exercise 5

Find the general solution for each equation in Finding General Solutions.

$\dfrac{dx}{dt} = x^2 - 6x + 14$

2.5

Exercise 6

Find the general solution for each equation in Finding General Solutions.

$y' = \dfrac{x^2}{y(2 + x^3)}$

2.5

Exercise 7

Find the general solution for each equation in Finding General Solutions.

$\dfrac{dy}{dx} = \dfrac{\sin x}{y}$

2.5

Exercise 8

Find the general solution for each equation in Finding General Solutions.

$xy' = \sqrt{1 - y^2}$

2.5

Exercise 9

Find the general solution for each equation in Finding General Solutions.

$\dfrac{dy}{dt} = \dfrac{4t - t^3}{4 + y^3}$

2.5

Exercise 10

Find the general solution for each equation in Finding General Solutions.

$y' = \dfrac{1}{ty + 2t + y + 2}$

2.5

Exercise 11

Find the general solution for each equation in Finding General Solutions.

$\dfrac{dx}{dt} = xe^t + 3e^t + x + 3$

2.5

Exercise 12

Solve the initial value problems in Solving Initial Value Problems.

$x \, dx - y^2 \, dy = 0$ , $y(0) = 1$

2.5

Exercise 13

Solve the initial value problems in Solving Initial Value Problems.

$\dfrac{dy}{dx} = \dfrac{y}{x}$ , $y(1) = -2$

2.5

Exercise 14

Solve the initial value problems in Solving Initial Value Problems.

$(t^2 + 1) \, dt + (x^2 + x) \, dx = 0$ , $x(0) = -1$

2.5

Exercise 15

Solve the initial value problems in Solving Initial Value Problems.

$\dfrac{dx}{dt} = x^2 - 6x + 14$ , $x(0) = 3$

2.5

Exercise 16

Solve the initial value problems in Solving Initial Value Problems.

$y' = \dfrac{x^2}{y(2 + x^3)}$ , $y(0) = -2$

2.5

Exercise 17

Solve the initial value problems in Solving Initial Value Problems.

$\dfrac{dy}{dx} = \dfrac{\sin x}{y}$ , $y(0) = 2$

2.5

Exercise 18

Solve the initial value problems in Solving Initial Value Problems.

$xy' = \sqrt{1 - y^2}$ , $y(1) = 0$

2.5

Exercise 19

Solve the initial value problems in Solving Initial Value Problems.

$\dfrac{dy}{dt} = \dfrac{4t - t^3}{4 + y^3}$ , $y(0) = 1$

2.5

Exercise 20

Solve the initial value problems in Solving Initial Value Problems.

$y' = \dfrac{1}{ty + 2t + y + 2}$ , $y(0) = -3$

2.5

Exercise 21

Solve the initial value problems in Solving Initial Value Problems.

$\dfrac{dx}{dt} = xe^t + 3e^t + x + 3$ , $x(0) = 0$

2.5

Exercise 23

A first-order differential equation, $y' = f(x, y)$ , is homogeneous if $f(x, y) = f(tx, ty)$ .

Show that the equation is homogeneous.
Let $y = xv$ and show that the equation in part (a) can be written as Use the fact that this new equation is separable to solve for $y$ .
Show that any homogeneous equation $y' = f(x, y)$ can be transformed into a separable differential equation by making the substitution $y = vx$ .
A function $f$ is said to homogeneous of degree $n$ i…

2.5

Exercise 24

Mr. Ratchett, an elderly American, was found murdered in his train compartment on the Orient Express at 7 AM. When his body was discovered, the famous detective Hercule Poirot noted that Ratchett had a body temperature of 28 degrees. The body had cooled to a temperature of 27 degrees one hour later. If the normal temperature of a human being is 37 degrees and the air temperature in the train is 22…

2.5

Exercise 25

You are starting a new position, and your employer has a generous retirement plan. If you put $500 a month into a 401(k) plan, your employer will match your contributions.

Assume that you are 25 years old and plan to retire at age 65, how large can you expect your 401(k) pension to be when you retire? Assume that your 401(k) plan will collect interest at a rate of 5%.
If you begin withdrawing $60,00…

2.5