First-Order Linear Equations

Multiplying y' + p(x) y = g(x) by an integrating factor so the left side collapses to a single derivative that integrates directly.

The ODE Project, First-Order Linear Equations

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24 exercises

Exercise 1

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

$\dfrac{dy}{dx} + 5y = 0$

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Exercise 2

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

$x' - 7x = 0$

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Exercise 3

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

$y' + 2xy = 0$

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Exercise 4

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

$\dfrac{dx}{dt} -t^2 x = 0$

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Exercise 5

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

$y' + \dfrac{2}{x}y = 0$

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Exercise 6

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

$\dfrac{dy}{dx} - 5y = e^x$

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Exercise 7

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

$y' - 5y = 10x$

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Exercise 8

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

$\dfrac{dx}{dt} - 5x = \sin 2t$

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Exercise 9

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

$y' - \dfrac{3}{x^2} y = \dfrac{1}{x^2}$

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Exercise 10

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

$\dfrac{dy}{dx} + \dfrac{2}{x} y = \dfrac{\sin x}{x^2}$

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Exercise 11

Solve the initial value problems in Solving Initial Value Problems.

$\dfrac{dy}{dx} + 5y = 0$ , $y(0) = 2$

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Exercise 12

Solve the initial value problems in Solving Initial Value Problems.

$x' - 7x = 0$ , $x(0) = 1$

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Exercise 13

Solve the initial value problems in Solving Initial Value Problems.

$y' + 2xy = 0$ , $y(0) = 3$

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Exercise 14

Solve the initial value problems in Solving Initial Value Problems.

$\dfrac{dx}{dt} -t^2 x = 0$ , $x(0) = -1$

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Exercise 15

Solve the initial value problems in Solving Initial Value Problems.

$y' + \dfrac{2}{x}y = 0$ , $y(1) = -3$

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Exercise 16

Solve the initial value problems in Solving Initial Value Problems.

$y' = - \dfrac{2y}{x + 1} + 2x$ , $y(0) = 2$

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Exercise 17

Solve the initial value problems in Solving Initial Value Problems.

$y' = - \dfrac{2y}{1 + x} + e^x$ , $y(0) = 1$

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Exercise 18

Solve the initial value problems in Solving Initial Value Problems.

$\dfrac{dx}{dt} - 5x = \sin 2t$ , $x(0) = \pi$

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Exercise 19

Solve the initial value problems in Solving Initial Value Problems.

$y' = y \tan x + \dfrac{e^x}{\cos x}$ , $y(0) = 1$

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Exercise 20

Solve the initial value problems in Solving Initial Value Problems.

$y' = - \dfrac{y}{x + 2} + \dfrac{\cos x}{x + 2}$ , $y(0) = 1$

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Exercise 21

A 600-liter tank initially contains 200 liters of water containing 10 kilograms of salt. Supposed that water containing 0.1 kilograms of salt per liter flows into the top of the tank at a rate of 10 liters per minute. The water in the tank is kept well mixed, and 5 liters per minute are removed from the bottom of the tank. How much salt is in the tank when the tank is full?

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Exercise 22

A manager in a communications company contributes $2400 per year into her retirement fund by making many small deposits throughout the year. The fund grows at a rate of 3.5% per year compounded continuously. After 35 years, she retires and begins and begins withdrawing from the retirement fund at a rate of $3500 per month. If she does not make any deposits after she retires, how long will her reti…

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Exercise 27

Bernoulli's equation is an equation of the form where $n \neq 0$ or 1. Bernoulli's equation is nonlinear and cannot be solved by the techniques that we have used to solve first order linear equations.

Using the substitution $z = y^{1- n}$ , show that we can transform Bernoulli's equation into the linear equation
Solve the equation $xy' + y = x^4y^3$ .

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Exercise 30

The differential equation is not autonomous, separable, or linear; however, we can solve this equation with a change of variable.

Transform this equation into a new differential equation of the form by letting $u = y - 4t$ .
Sketch the phase line for this new equation, $u' = f(u)$ , and sketch several solutions.
Find the solutions of the original differential equation that correspond to the equilibrium solutions of $u' = f(u)$ . Gr…

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