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17Calculus - Differential Equations - First Order Practice

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Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
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This page contains a mixed bag of practice problems solving first order differential equations based on a video from one of our favorite instructors. We have laid out each practice problem and included the video clip containing each solution.
We recommend that you download this pdf of his list of problems before starting. However, his videos do not follow this pdf exactly.

Make sure you support the guy that did this video. He put a LOT of work into, not just doing the video, but also preparing the problems and making sure his solutions were correct. He did a GREAT job. So go to YouTube and like this video and follow him. He is one of our favorite instructors. (By the way, we are not receiving any compensation from him. We just think his videos will help you.) Here is his YouTube channel link.

Practice

Find the general solution to these differential equations. If initial conditions are given, find the particular solution as well.

Basic

\( (\sin(y) - y\sin(x))dx + (\cos(x)+x\cos(y))dy = 0 \)

Problem Statement

\( (\sin(y) - y\sin(x))dx + (\cos(x)+x\cos(y))dy = 0 \)

Solution

blackpenredpen - 3664 video solution

video by blackpenredpen

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\( (4x^2-10y)dx + (2x^3/y-15x)dy = 0, y(2) = -1 \)

Problem Statement

\( (4x^2-10y)dx + (2x^3/y-15x)dy = 0, y(2) = -1 \)

Hint

This is almost exact. Use the special integrating factor \(\mu(x,y) = x^m y^n\)

Problem Statement

\( (4x^2-10y)dx + (2x^3/y-15x)dy = 0, y(2) = -1 \)

Hint

This is almost exact. Use the special integrating factor \(\mu(x,y) = x^m y^n\)

Solution

blackpenredpen - 3665 video solution

video by blackpenredpen

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\( dy/dx = \cos(x) - y\sec(x), y(0) = 2 \)

Problem Statement

\( dy/dx = \cos(x) - y\sec(x), y(0) = 2 \)

Solution

blackpenredpen - 3666 video solution

video by blackpenredpen

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\( (xy + y^2 + x^2)dx - x^2dy = 0 \)

Problem Statement

\( (xy + y^2 + x^2)dx - x^2dy = 0 \)

Solution

blackpenredpen - 3667 video solution

video by blackpenredpen

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\( (2x+\cos y)dx + (x^2 + \tan y + \cos^2 y)dy = 0 \)

Problem Statement

\( (2x+\cos y)dx + (x^2 + \tan y + \cos^2 y)dy = 0 \)

Solution

blackpenredpen - 3668 video solution

video by blackpenredpen

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\(\displaystyle{ (x^2+1)\frac{dy}{dx} + xy - x = 0 }\)

Problem Statement

\(\displaystyle{ (x^2+1)\frac{dy}{dx} + xy - x = 0 }\)

Solution

blackpenredpen - 3669 video solution

video by blackpenredpen

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\( dy/dx = 2-\sqrt{ 2x-y+3 } \)

Problem Statement

\( dy/dx = 2-\sqrt{ 2x-y+3 } \)

Solution

blackpenredpen - 3670 video solution

video by blackpenredpen

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\( 2xye^{x^2} dx + (e^{x^2}- 1/y)dy = 0 \)

Problem Statement

\( 2xye^{x^2} dx + (e^{x^2}- 1/y)dy = 0 \)

Solution

blackpenredpen - 3671 video solution

video by blackpenredpen

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\( e^y dx - e^x dy = 0, y(0) = 0 \)

Problem Statement

\( e^y dx - e^x dy = 0, y(0) = 0 \)

Solution

blackpenredpen - 3672 video solution

video by blackpenredpen

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\( dy/dx = -y(1+xy^2), y(0) = 1/2 \)

Problem Statement

\( dy/dx = -y(1+xy^2), y(0) = 1/2 \)

Solution

blackpenredpen - 3673 video solution

video by blackpenredpen

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\( (y^2-x^2)dx - xydy = 0, y(1) = 3 \)

Problem Statement

\( (y^2-x^2)dx - xydy = 0, y(1) = 3 \)

Solution

He works this several different ways. Make sure to watch the complete video so that you can learn how to solve this in more ways than one.

blackpenredpen - 3674 video solution

video by blackpenredpen

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\( dP/dt = P(a-b\ln P), P(0) = P_0 \)

Problem Statement

\( dP/dt = P(a-b\ln P), P(0) = P_0 \)

Solution

blackpenredpen - 3675 video solution

video by blackpenredpen

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\( dP/dt = kP(1-P/M), P(0) = P_0 \)

Problem Statement

\( dP/dt = kP(1-P/M), P(0) = P_0 \)

Solution

blackpenredpen - 3678 video solution

video by blackpenredpen

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\(\displaystyle{ \frac{dy}{dx} = \frac{x-y}{x+y} }\)

Problem Statement

Find the general solution to the differential equation \(\displaystyle{ \frac{dy}{dx} = \frac{x-y}{x+y} }\)

Solution

Here are two videos by two different instructors with similar solutions to this differential equation.

SyberMath - 3679 video solution

video by SyberMath

blackpenredpen - 3679 video solution

video by blackpenredpen

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\( dy/dx= \sqrt{y} - y \)

Problem Statement

\( dy/dx= \sqrt{y} - y \)

Solution

3680 video solution

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\(\displaystyle{ dy/dx = \frac{1-2y-\sin^2(x^2y)}{x} }\)

Problem Statement

\(\displaystyle{ dy/dx = \frac{1-2y-\sin^2(x^2y)}{x} }\)

Solution

blackpenredpen - 3681 video solution

video by blackpenredpen

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\( (-3y+ x^2y^2)dx + (x-2x^3y+x^4/y)dy = 0 \)

Problem Statement

\( (-3y+ x^2y^2)dx + (x-2x^3y+x^4/y)dy = 0 \)

Solution

blackpenredpen - 3682 video solution

video by blackpenredpen

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\( 2xye^{x^2}dx + (e^{x^2}-1/y)dy = 0 \)

Problem Statement

\( 2xye^{x^2}dx + (e^{x^2}-1/y)dy = 0 \)

Solution

blackpenredpen - 3684 video solution

video by blackpenredpen

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\(\displaystyle{ y' = -x + \frac{1}{2x}y, y(1) = 0 }\)

Problem Statement

\(\displaystyle{ y' = -x + \frac{1}{2x}y, y(1) = 0 }\)

Solution

blackpenredpen - 3685 video solution

video by blackpenredpen

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\( y' = x^2 + 2xy + y^2, y(1) = 0 \)

Problem Statement

\( y' = x^2 + 2xy + y^2, y(1) = 0 \)

Solution

blackpenredpen - 3687 video solution

video by blackpenredpen

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Intermediate

\(\displaystyle{ dy/dx = \frac{xy^2-\sin x \cos x}{y(1-x^2)}, y(0) = 4 }\)

Problem Statement

\(\displaystyle{ dy/dx = \frac{xy^2-\sin x \cos x}{y(1-x^2)}, y(0) = 4 }\)

Solution

blackpenredpen - 3677 video solution

video by blackpenredpen

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\(\displaystyle{ y' = -x + \frac{1}{2x}y + y^2, y(1) = 0 }\)

Problem Statement

\(\displaystyle{ y' = -x + \frac{1}{2x}y + y^2, y(1) = 0 }\)

Hint

Start with \( y_1 = x^n\) and find \( y = y_1 + v \)

Problem Statement

\(\displaystyle{ y' = -x + \frac{1}{2x}y + y^2, y(1) = 0 }\)

Hint

Start with \( y_1 = x^n\) and find \( y = y_1 + v \)

Solution

blackpenredpen - 3686 video solution

video by blackpenredpen

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\( y = \left( \dfrac{dy}{dx} \right)^2 \)

Problem Statement

Solve the differential equation \( y = \left( \dfrac{dy}{dx} \right)^2 \)

Solution

SyberMath - 4346 video solution

video by SyberMath

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Advanced

\( y = xy' + \sqrt{(y')^2 + 1} \)

Problem Statement

\( y = xy' + \sqrt{(y')^2 + 1} \)

Hint

Start by taking the derivative of the original differential equation.

Problem Statement

\( y = xy' + \sqrt{(y')^2 + 1} \)

Hint

Start by taking the derivative of the original differential equation.

Solution

blackpenredpen - 3676 video solution

video by blackpenredpen

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\( y=xy'-e^{y'} \)

Problem Statement

\( y=xy'-e^{y'} \)

Solution

blackpenredpen - 3683 video solution

video by blackpenredpen

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Really UNDERSTAND Differential Equations

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Topics You Need To Understand For This Page

basics of differential equations

all first order techniques

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Practice Instructions

Find the general solution to these differential equations. If initial conditions are given, find the particular solution as well.

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