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Separable differential equations practice problems

Separate the variables, integrate both sides, and use the initial condition. 70 problems with worked solutions; in 70 of them every equation is proved by a computer algebra system.

10 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 6.5

medium Video
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 2 y,\ y(0) = 4 \).
  1. Separate the variables, putting every y on one side and every x on the other, and integrate both sides.
    Reviewed
  2. Use y(0) to find the constant of integration, then solve for y.
    Reviewed
  3. \[ \frac{d}{d x} 4 e^{2 x} = 8 e^{2 x} \]
    The solution satisfies the equation.✓ Proved
  4. \[ 4 \]
    And the initial condition.Reviewed
Answer \( y = 4 e^{2 x} \)
The slope field of y' = 2y: short segments showing the slope at each point. Several solution curves follow the field; the one through (0, 4) is highlighted, y = 4e^(2x).

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.12

medium Video
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - x y,\ y(0) = 5 \).
  1. Separate the variables, putting every y on one side and every x on the other, and integrate both sides.
    Reviewed
  2. Use y(0) to find the constant of integration, then solve for y.
    Reviewed
  3. \[ \frac{d}{d x} 5 e^{- \frac{x^{2}}{2}} = - 5 x e^{- \frac{x^{2}}{2}} \]
    The solution satisfies the equation.✓ Proved
  4. \[ 5 \]
    And the initial condition.Reviewed
Answer \( y = 5 e^{- \frac{x^{2}}{2}} \)
The slope field of y' = -x·y: short segments showing the slope at each point. Several solution curves follow the field; the one through (0, 5) is highlighted, y = 5e^(-x²/2).

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.13

medium Video
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 3 x y,\ y(0) = 5 \).
  1. Separate the variables, putting every y on one side and every x on the other, and integrate both sides.
    Reviewed
  2. Use y(0) to find the constant of integration, then solve for y.
    Reviewed
  3. \[ \frac{d}{d x} 5 e^{- \frac{3 x^{2}}{2}} = - 15 x e^{- \frac{3 x^{2}}{2}} \]
    The solution satisfies the equation.✓ Proved
  4. \[ 5 \]
    And the initial condition.Reviewed
Answer \( y = 5 e^{- \frac{3 x^{2}}{2}} \)
The slope field of y' = -3x·y: short segments showing the slope at each point. Several solution curves follow the field; the one through (0, 5) is highlighted, y = 5e^(-3x²/2).

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.52

medium Video
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 2 x y,\ y(0) = 3 \).
  1. Separate the variables, putting every y on one side and every x on the other, and integrate both sides.
    Reviewed
  2. Use y(0) to find the constant of integration, then solve for y.
    Reviewed
  3. \[ \frac{d}{d x} 3 e^{x^{2}} = 6 x e^{x^{2}} \]
    The solution satisfies the equation.✓ Proved
  4. \[ 3 \]
    And the initial condition.Reviewed
Answer \( y = 3 e^{x^{2}} \)
The slope field of y' = 2x·y: short segments showing the slope at each point. Several solution curves follow the field; the one through (0, 3) is highlighted, y = 3e^(x²).

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.53

medium Video
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 2 y,\ y(0) = 5 \).
  1. Separate the variables, putting every y on one side and every x on the other, and integrate both sides.
    Reviewed
  2. Use y(0) to find the constant of integration, then solve for y.
    Reviewed
  3. \[ \frac{d}{d x} 5 e^{- 2 x} = - 10 e^{- 2 x} \]
    The solution satisfies the equation.✓ Proved
  4. \[ 5 \]
    And the initial condition.Reviewed
Answer \( y = 5 e^{- 2 x} \)
The slope field of y' = -2y: short segments showing the slope at each point. Several solution curves follow the field; the one through (0, 5) is highlighted, y = 5e^(-2x).

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.121

medium Video
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = x y,\ y(0) = 1 \).
  1. Separate the variables, putting every y on one side and every x on the other, and integrate both sides.
    Reviewed
  2. Use y(0) to find the constant of integration, then solve for y.
    Reviewed
  3. \[ \frac{d}{d x} e^{\frac{x^{2}}{2}} = x e^{\frac{x^{2}}{2}} \]
    The solution satisfies the equation.✓ Proved
  4. \[ 1 \]
    And the initial condition.Reviewed
Answer \( y = e^{\frac{x^{2}}{2}} \)
The slope field of y' = x·y: short segments showing the slope at each point. Several solution curves follow the field; the one through (0, 1) is highlighted, y = e^(x²/2).

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.188

medium Video
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{3 x}{y},\ y(0) = 2 \).
  1. Separate the variables, putting every y on one side and every x on the other, and integrate both sides.
    Reviewed
  2. Use y(0) to find the constant of integration, then solve for y.
    Reviewed
  3. \[ \frac{d}{d x} \sqrt{3 x^{2} + 4} = \frac{3 x}{\sqrt{3 x^{2} + 4}} \]
    The solution satisfies the equation.✓ Proved
  4. \[ 2 \]
    And the initial condition.Reviewed
Answer \( y = \sqrt{3 x^{2} + 4} \)
The slope field of y' = 3x/y: short segments showing the slope at each point. Several solution curves follow the field; the one through (0, 2) is highlighted, y = sqrt(3x² + 4).

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.1

easy Video
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - y,\ y(0) = 5 \).
  1. Separate the variables, putting every y on one side and every x on the other, and integrate both sides.
    Reviewed
  2. Use y(0) to find the constant of integration, then solve for y.
    Reviewed
  3. \[ \frac{d}{d x} 5 e^{- x} = - 5 e^{- x} \]
    The solution satisfies the equation.✓ Proved
  4. \[ 5 \]
    And the initial condition.Reviewed
Answer \( y = 5 e^{- x} \)
The slope field of y' = -y: short segments showing the slope at each point. Several solution curves follow the field; the one through (0, 5) is highlighted, y = 5e^(-x).

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.2

easy Video
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = y^{2},\ y(0) = \frac{1}{4} \).
  1. Separate the variables, putting every y on one side and every x on the other, and integrate both sides.
    Reviewed
  2. Use y(0) to find the constant of integration, then solve for y.
    Reviewed
  3. \[ \frac{d}{d x} \frac{1}{4 - x} = \frac{1}{\left(4 - x\right)^{2}} \]
    The solution satisfies the equation.✓ Proved
  4. \[ 1 \cdot \frac{1}{4} = \frac{1}{4} \]
    And the initial condition.Reviewed
Answer \( y = \frac{1}{4 - x} \)
The slope field of y' = y²: short segments showing the slope at each point. Several solution curves follow the field; the one through (0, 1/4) is highlighted, y = 1/(4 - x).

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.14

easy Video
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = y,\ y(0) = 4 \).
  1. Separate the variables, putting every y on one side and every x on the other, and integrate both sides.
    Reviewed
  2. Use y(0) to find the constant of integration, then solve for y.
    Reviewed
  3. \[ \frac{d}{d x} 4 e^{x} = 4 e^{x} \]
    The solution satisfies the equation.✓ Proved
  4. \[ 4 \]
    And the initial condition.Reviewed
Answer \( y = 4 e^{x} \)
The slope field of y' = y: short segments showing the slope at each point. Several solution curves follow the field; the one through (0, 4) is highlighted, y = 4e^(x).

✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More separable differential equations practice

Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - y,\ y(0) = 2 \).
Problem 6.3easy Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = y^{2},\ y(0) = \frac{1}{3} \).
Problem 6.7easy Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = y^{2},\ y(0) = 1 \).
Problem 6.10easy Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = y^{2},\ y(0) = \frac{1}{2} \).
Problem 6.15easy Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = y,\ y(0) = 5 \).
Problem 6.60easy Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = y,\ y(0) = 1 \).
Problem 6.139easy Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 3 y,\ y(0) = 1 \).
Problem 6.140easy Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - y,\ y(0) = 3 \).
Problem 6.142easy Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = y,\ y(0) = 3 \).
Problem 6.164easy Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - y,\ y(0) = 1 \).
Problem 6.168easy Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 2 y,\ y(0) = 1 \).
Problem 6.189easy Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 2 x y,\ y(0) = 1 \).
Problem 6.194easy Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{3 x}{y},\ y(0) = 3 \).
Problem 6.4medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{3 x}{y},\ y(0) = 5 \).
Problem 6.6medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{2 x}{y},\ y(0) = 5 \).
Problem 6.8medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{3 x}{y},\ y(0) = 4 \).
Problem 6.9medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 2 y,\ y(0) = 3 \).
Problem 6.11medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{x}{y},\ y(0) = 3 \).
Problem 6.46medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 2 x y,\ y(0) = 2 \).
Problem 6.47medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{x}{y},\ y(0) = 1 \).
Problem 6.48medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{2 x}{y},\ y(0) = 2 \).
Problem 6.49medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 3 x y,\ y(0) = 4 \).
Problem 6.50medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{3 x}{y},\ y(0) = 1 \).
Problem 6.51medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 3 x y,\ y(0) = 2 \).
Problem 6.54medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - x y,\ y(0) = 2 \).
Problem 6.55medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 3 x y,\ y(0) = 5 \).
Problem 6.56medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 2 y,\ y(0) = 4 \).
Problem 6.57medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 3 x y,\ y(0) = 3 \).
Problem 6.58medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 2 x y,\ y(0) = 5 \).
Problem 6.59medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 3 y,\ y(0) = 4 \).
Problem 6.91medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 3 y,\ y(0) = 5 \).
Problem 6.92medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{2 x}{y},\ y(0) = 4 \).
Problem 6.93medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{2 x}{y},\ y(0) = 1 \).
Problem 6.94medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 3 y,\ y(0) = 3 \).
Problem 6.95medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{x}{y},\ y(0) = 5 \).
Problem 6.96medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 2 x y,\ y(0) = 1 \).
Problem 6.97medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{x}{y},\ y(0) = 2 \).
Problem 6.98medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - x y,\ y(0) = 4 \).
Problem 6.115medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 2 x y,\ y(0) = 5 \).
Problem 6.116medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 3 x y,\ y(0) = 2 \).
Problem 6.117medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{2 x}{y},\ y(0) = 3 \).
Problem 6.118medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 2 x y,\ y(0) = 3 \).
Problem 6.119medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 3 y,\ y(0) = 4 \).
Problem 6.120medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 3 y,\ y(0) = 2 \).
Problem 6.122medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 2 y,\ y(0) = 5 \).
Problem 6.141medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = x y,\ y(0) = 2 \).
Problem 6.143medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 2 x y,\ y(0) = 4 \).
Problem 6.144medium Video✓ Every equation proved
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 3 y,\ y(0) = 5 \).
Problem 6.145medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 2 y,\ y(0) = 2 \).
Problem 6.146medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 2 y,\ y(0) = 3 \).
Problem 6.163medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 2 y,\ y(0) = 2 \).
Problem 6.165medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - x y,\ y(0) = 3 \).
Problem 6.166medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 2 y,\ y(0) = 1 \).
Problem 6.167medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 3 y,\ y(0) = 3 \).
Problem 6.169medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = x y,\ y(0) = 3 \).
Problem 6.170medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = - 3 y,\ y(0) = 1 \).
Problem 6.187medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 2 x y,\ y(0) = 4 \).
Problem 6.190medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = x y,\ y(0) = 5 \).
Problem 6.191medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = 3 y,\ y(0) = 2 \).
Problem 6.192medium Video✓ Nihil obstat
Solve the initial-value problem \( \displaystyle \dfrac{dy}{dx} = \frac{x}{y},\ y(0) = 4 \).
Problem 6.193medium Video✓ Every equation proved