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Differential equations by direct integration practice problems

Solve y′ = f(x) or y″ = f(x) by integrating, with or without an initial condition. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

20 worked problems

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

Problem 6.428

medium
Find the general solution of \( \displaystyle y'' = - 2 x + \cos{\left(x \right)} \).
  1. \[ \frac{d}{d x} \left(- x^{2} + \sin{\left(x \right)}\right) = - 2 x + \cos{\left(x \right)} \]
    Integrate once: y′ = -x**2 + sin(x) + C₁.✓ Proved
  2. \[ \frac{d}{d x} \left(- \frac{x^{3}}{3} - \cos{\left(x \right)}\right) = - x^{2} + \sin{\left(x \right)} \]
    Integrate again: y = -x**3/3 - cos(x) + C₁x + C₂.✓ Proved
Answer \( y = C_{1} x + C_{2} - \frac{x^{3}}{3} - \cos{\left(x \right)} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.430

medium
Find the general solution of \( \displaystyle y'' = 6 x \).
  1. \[ \frac{d}{d x} 3 x^{2} = 6 x \]
    Integrate once: y′ = 3*x**2 + C₁.✓ Proved
  2. \[ \frac{d}{d x} x^{3} = 3 x^{2} \]
    Integrate again: y = x**3 + C₁x + C₂.✓ Proved
Answer \( y = C_{1} x + C_{2} + x^{3} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.431

medium
Find the general solution of \( \displaystyle y'' = \sin{\left(2 x \right)} \).
  1. \[ \frac{d}{d x} \left(- \frac{\cos{\left(2 x \right)}}{2}\right) = \sin{\left(2 x \right)} \]
    Integrate once: y′ = -cos(2*x)/2 + C₁.✓ Proved
  2. \[ \frac{d}{d x} \left(- \frac{\sin{\left(2 x \right)}}{4}\right) = - \frac{\cos{\left(2 x \right)}}{2} \]
    Integrate again: y = -sin(2*x)/4 + C₁x + C₂.✓ Proved
Answer \( y = C_{1} x + C_{2} - \frac{\sin{\left(2 x \right)}}{4} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.433

medium
Find the general solution of \( \displaystyle y'' = x^{2} - 4 \).
  1. \[ \frac{d}{d x} \left(\frac{x^{3}}{3} - 4 x\right) = x^{2} - 4 \]
    Integrate once: y′ = x**3/3 - 4*x + C₁.✓ Proved
  2. \[ \frac{d}{d x} \left(\frac{x^{4}}{12} - 2 x^{2}\right) = \frac{x^{3}}{3} - 4 x \]
    Integrate again: y = x**4/12 - 2*x**2 + C₁x + C₂.✓ Proved
Answer \( y = C_{1} x + C_{2} + \frac{x^{4}}{12} - 2 x^{2} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.573

medium
Find the general solution of \( \displaystyle y'' = \frac{1}{x} \).
  1. \[ \frac{d}{d x} \ln{\left(x \right)} = \frac{1}{x} \]
    Integrate once: y′ = log(x) + C₁.✓ Proved
  2. \[ \frac{d}{d x} \left(x \ln{\left(x \right)} - x\right) = \ln{\left(x \right)} \]
    Integrate again: y = x*log(x) - x + C₁x + C₂.✓ Proved
Answer \( y = C_{1} x + C_{2} + x \ln{\left(x \right)} - x \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.577

medium
Find the general solution of \( \displaystyle y'' = 3 x + e^{x} \).
  1. \[ \frac{d}{d x} \left(\frac{3 x^{2}}{2} + e^{x}\right) = 3 x + e^{x} \]
    Integrate once: y′ = 3*x**2/2 + exp(x) + C₁.✓ Proved
  2. \[ \frac{d}{d x} \left(\frac{x^{3}}{2} + e^{x}\right) = \frac{3 x^{2}}{2} + e^{x} \]
    Integrate again: y = x**3/2 + exp(x) + C₁x + C₂.✓ Proved
Answer \( y = C_{1} x + C_{2} + \frac{x^{3}}{2} + e^{x} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.685

medium
Find the general solution of \( \displaystyle y'' = 2 x e^{x^{2}} \).
  1. \[ \frac{d}{d x} e^{x^{2}} = 2 x e^{x^{2}} \]
    Integrate once: y′ = exp(x**2) + C₁.✓ Proved
  2. \[ \frac{d}{d x} \frac{\sqrt{\pi} \operatorname{erfi}{\left(x \right)}}{2} = e^{x^{2}} \]
    Integrate again: y = sqrt(pi)*erfi(x)/2 + C₁x + C₂.✓ Proved
Answer \( y = C_{1} x + C_{2} + \frac{\sqrt{\pi} \operatorname{erfi}{\left(x \right)}}{2} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.426

easy
Solve \( \displaystyle y' = 2 x e^{x^{2}} \) with \( \displaystyle y(0) = -1 \).
  1. \[ \frac{d}{d x} e^{x^{2}} = 2 x e^{x^{2}} \]
    Integrate: y = this + C.✓ Proved
  2. \[ - \left. e^{x^{2}} \right|_{\substack{ x=0 }} - 1 = -2 \]
    y(0) = -1 fixes C.✓ Proved
  3. \[ \left. e^{x^{2}} - 2 \right|_{\substack{ x=0 }} = -1 \]
    The initial condition holds.✓ Proved
Answer \( y = e^{x^{2}} - 2 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.432

easy
Solve \( \displaystyle y' = - 2 x + \cos{\left(x \right)} \) with \( \displaystyle y(0) = 10 \).
  1. \[ \frac{d}{d x} \left(- x^{2} + \sin{\left(x \right)}\right) = - 2 x + \cos{\left(x \right)} \]
    Integrate: y = this + C.✓ Proved
  2. \[ 10 - \left. - x^{2} + \sin{\left(x \right)} \right|_{\substack{ x=0 }} = 10 \]
    y(0) = 10 fixes C.✓ Proved
  3. \[ \left. - x^{2} + \sin{\left(x \right)} + 10 \right|_{\substack{ x=0 }} = 10 \]
    The initial condition holds.✓ Proved
Answer \( y = - x^{2} + \sin{\left(x \right)} + 10 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.436

easy
Solve \( \displaystyle y' = \sin{\left(2 x \right)} \) with \( \displaystyle y(1) = -4 \).
  1. \[ \frac{d}{d x} \left(- \frac{\cos{\left(2 x \right)}}{2}\right) = \sin{\left(2 x \right)} \]
    Integrate: y = this + C.✓ Proved
  2. \[ - \left. \frac{\left(-1\right) \cos{\left(2 x \right)}}{2} \right|_{\substack{ x=1 }} - 4 = -4 + \frac{\cos{\left(2 \right)}}{2} \]
    y(1) = -4 fixes C.✓ Proved
  3. \[ \left. \frac{\left(-1\right) \cos{\left(2 x \right)}}{2} - 4 + \frac{\cos{\left(2 \right)}}{2} \right|_{\substack{ x=1 }} = -4 \]
    The initial condition holds.✓ Proved
Answer \( y = - \frac{\cos{\left(2 x \right)}}{2} - 4 + \frac{\cos{\left(2 \right)}}{2} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.437

easy
Solve \( \displaystyle y' = \sin{\left(2 x \right)} \) with \( \displaystyle y(0) = 0 \).
  1. \[ \frac{d}{d x} \left(- \frac{\cos{\left(2 x \right)}}{2}\right) = \sin{\left(2 x \right)} \]
    Integrate: y = this + C.✓ Proved
  2. \[ - \left. \frac{\left(-1\right) \cos{\left(2 x \right)}}{2} \right|_{\substack{ x=0 }} + 0 = \frac{1}{2} \]
    y(0) = 0 fixes C.✓ Proved
  3. \[ \left. - \frac{\cos{\left(2 x \right)}}{2} + 1 \cdot \frac{1}{2} \right|_{\substack{ x=0 }} = 0 \]
    The initial condition holds.✓ Proved
Answer \( y = \frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.440

easy
Solve \( \displaystyle y' = x^{2} + 2 \) with \( \displaystyle y(1) = 8 \).
  1. \[ \frac{d}{d x} \left(\frac{x^{3}}{3} + 2 x\right) = x^{2} + 2 \]
    Integrate: y = this + C.✓ Proved
  2. \[ 8 - \left. \frac{x^{3}}{3} + 2 x \right|_{\substack{ x=1 }} = \frac{17}{3} \]
    y(1) = 8 fixes C.✓ Proved
  3. \[ \left. \frac{x^{3}}{3} + 2 x + \frac{17}{3} \right|_{\substack{ x=1 }} = 8 \]
    The initial condition holds.✓ Proved
Answer \( y = \frac{x^{3}}{3} + 2 x + \frac{17}{3} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.530

easy
Solve \( \displaystyle y' = 3 x + e^{x} \) with \( \displaystyle y(0) = 7 \).
  1. \[ \frac{d}{d x} \left(\frac{3 x^{2}}{2} + e^{x}\right) = 3 x + e^{x} \]
    Integrate: y = this + C.✓ Proved
  2. \[ 7 - \left. \frac{3 x^{2}}{2} + e^{x} \right|_{\substack{ x=0 }} = 6 \]
    y(0) = 7 fixes C.✓ Proved
  3. \[ \left. \frac{3 x^{2}}{2} + e^{x} + 6 \right|_{\substack{ x=0 }} = 7 \]
    The initial condition holds.✓ Proved
Answer \( y = \frac{3 x^{2}}{2} + e^{x} + 6 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.571

easy
Solve \( \displaystyle y' = 2 x e^{x^{2}} \) with \( \displaystyle y(0) = 8 \).
  1. \[ \frac{d}{d x} e^{x^{2}} = 2 x e^{x^{2}} \]
    Integrate: y = this + C.✓ Proved
  2. \[ 8 - \left. e^{x^{2}} \right|_{\substack{ x=0 }} = 7 \]
    y(0) = 8 fixes C.✓ Proved
  3. \[ \left. e^{x^{2}} + 7 \right|_{\substack{ x=0 }} = 8 \]
    The initial condition holds.✓ Proved
Answer \( y = e^{x^{2}} + 7 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.579

easy
Solve \( \displaystyle y' = x^{2} - 3 \) with \( \displaystyle y(1) = 7 \).
  1. \[ \frac{d}{d x} \left(\frac{x^{3}}{3} - 3 x\right) = x^{2} - 3 \]
    Integrate: y = this + C.✓ Proved
  2. \[ 7 - \left. \frac{x^{3}}{3} - 3 x \right|_{\substack{ x=1 }} = \frac{29}{3} \]
    y(1) = 7 fixes C.✓ Proved
  3. \[ \left. \frac{x^{3}}{3} - 3 x + \frac{29}{3} \right|_{\substack{ x=1 }} = 7 \]
    The initial condition holds.✓ Proved
Answer \( y = \frac{x^{3}}{3} - 3 x + \frac{29}{3} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.583

easy
Solve \( \displaystyle y' = 6 x \) with \( \displaystyle y(0) = 8 \).
  1. \[ \frac{d}{d x} 3 x^{2} = 6 x \]
    Integrate: y = this + C.✓ Proved
  2. \[ 8 - \left. 3 x^{2} \right|_{\substack{ x=0 }} = 8 \]
    y(0) = 8 fixes C.✓ Proved
  3. \[ \left. 3 x^{2} + 8 \right|_{\substack{ x=0 }} = 8 \]
    The initial condition holds.✓ Proved
Answer \( y = 3 x^{2} + 8 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.683

easy
Solve \( \displaystyle y' = x^{2} - 1 \) with \( \displaystyle y(0) = -5 \).
  1. \[ \frac{d}{d x} \left(\frac{x^{3}}{3} - x\right) = x^{2} - 1 \]
    Integrate: y = this + C.✓ Proved
  2. \[ - \left. \frac{x^{3}}{3} - x \right|_{\substack{ x=0 }} - 5 = -5 \]
    y(0) = -5 fixes C.✓ Proved
  3. \[ \left. \frac{x^{3}}{3} - x - 5 \right|_{\substack{ x=0 }} = -5 \]
    The initial condition holds.✓ Proved
Answer \( y = \frac{x^{3}}{3} - x - 5 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.688

easy
Solve \( \displaystyle y' = x^{2} + 3 \) with \( \displaystyle y(0) = -1 \).
  1. \[ \frac{d}{d x} \left(\frac{x^{3}}{3} + 3 x\right) = x^{2} + 3 \]
    Integrate: y = this + C.✓ Proved
  2. \[ - \left. \frac{x^{3}}{3} + 3 x \right|_{\substack{ x=0 }} - 1 = -1 \]
    y(0) = -1 fixes C.✓ Proved
  3. \[ \left. \frac{x^{3}}{3} + 3 x - 1 \right|_{\substack{ x=0 }} = -1 \]
    The initial condition holds.✓ Proved
Answer \( y = \frac{x^{3}}{3} + 3 x - 1 \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.434

easy
Find the general solution of \( \displaystyle y' = 3 x + e^{x} \).
  1. \[ \frac{d}{d x} \left(\frac{3 x^{2}}{2} + e^{x}\right) = 3 x + e^{x} \]
    Integrate both sides; every antiderivative differs by a constant C₁.✓ Proved
Answer \( y = C_{1} + \frac{3 x^{2}}{2} + e^{x} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 6.435

easy
Find the general solution of \( \displaystyle y' = 6 x \).
  1. \[ \frac{d}{d x} 3 x^{2} = 6 x \]
    Integrate both sides; every antiderivative differs by a constant C₁.✓ Proved
Answer \( y = C_{1} + 3 x^{2} \)

✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. 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 differential equations by direct integration practice

Solve \( \displaystyle y' = \frac{1}{x} \) with \( \displaystyle y(1) = 8 \).
Problem 6.427easy✓ Every equation proved
Solve \( \displaystyle y' = 6 x \) with \( \displaystyle y(1) = -5 \).
Problem 6.429easy✓ Every equation proved
Solve \( \displaystyle y' = \sin{\left(2 x \right)} \) with \( \displaystyle y(0) = -4 \).
Problem 6.438easy✓ Nihil obstat
Solve \( \displaystyle y' = \sin{\left(2 x \right)} \) with \( \displaystyle y(1) = 5 \).
Problem 6.439easy✓ Nihil obstat
Solve \( \displaystyle y' = \frac{1}{x} \) with \( \displaystyle y(1) = 6 \).
Problem 6.526easy✓ Every equation proved
Solve \( \displaystyle y' = - 2 x + \cos{\left(x \right)} \) with \( \displaystyle y(1) = 5 \).
Problem 6.527easy✓ Nihil obstat
Solve \( \displaystyle y' = \frac{1}{x} \) with \( \displaystyle y(1) = -3 \).
Problem 6.528easy✓ Every equation proved
Find the general solution of \( \displaystyle y' = - 2 x + \cos{\left(x \right)} \).
Problem 6.529easy✓ Nihil obstat
Solve \( \displaystyle y' = 3 x + e^{x} \) with \( \displaystyle y(1) = 1 \).
Problem 6.531easy✓ Nihil obstat
Solve \( \displaystyle y' = 3 x + e^{x} \) with \( \displaystyle y(0) = 3 \).
Problem 6.532easy✓ Nihil obstat
Solve \( \displaystyle y' = 2 x e^{x^{2}} \) with \( \displaystyle y(0) = -2 \).
Problem 6.533easy✓ Every equation proved
Solve \( \displaystyle y' = 6 x \) with \( \displaystyle y(0) = 9 \).
Problem 6.534easy✓ Every equation proved
Find the general solution of \( \displaystyle y' = x^{2} + 3 \).
Problem 6.535easy✓ Nihil obstat
Solve \( \displaystyle y' = 3 x + e^{x} \) with \( \displaystyle y(1) = 0 \).
Problem 6.572easy✓ Nihil obstat
Find the general solution of \( \displaystyle y' = 2 x e^{x^{2}} \).
Problem 6.574easy✓ Nihil obstat
Solve \( \displaystyle y' = \frac{1}{x} \) with \( \displaystyle y(1) = 10 \).
Problem 6.575easy✓ Every equation proved
Solve \( \displaystyle y' = 2 x e^{x^{2}} \) with \( \displaystyle y(1) = -1 \).
Problem 6.576easy✓ Nihil obstat
Solve \( \displaystyle y' = \sin{\left(2 x \right)} \) with \( \displaystyle y(0) = 4 \).
Problem 6.578easy✓ Nihil obstat
Solve \( \displaystyle y' = \sin{\left(2 x \right)} \) with \( \displaystyle y(0) = 5 \).
Problem 6.580easy✓ Nihil obstat
Solve \( \displaystyle y' = x^{2} - 4 \) with \( \displaystyle y(0) = 9 \).
Problem 6.581easy✓ Every equation proved
Solve \( \displaystyle y' = x^{2} - 2 \) with \( \displaystyle y(0) = 8 \).
Problem 6.582easy✓ Every equation proved
Find the general solution of \( \displaystyle y' = x^{2} - 4 \).
Problem 6.584easy✓ Nihil obstat
Solve \( \displaystyle y' = - 2 x + \cos{\left(x \right)} \) with \( \displaystyle y(1) = 4 \).
Problem 6.585easy✓ Nihil obstat
Find the general solution of \( \displaystyle y' = \sin{\left(2 x \right)} \).
Problem 6.679easy✓ Nihil obstat
Solve \( \displaystyle y' = \frac{1}{x} \) with \( \displaystyle y(1) = 5 \).
Problem 6.680easy✓ Every equation proved
Solve \( \displaystyle y' = - 2 x + \cos{\left(x \right)} \) with \( \displaystyle y(1) = 1 \).
Problem 6.681easy✓ Nihil obstat
Solve \( \displaystyle y' = \sin{\left(2 x \right)} \) with \( \displaystyle y(0) = 8 \).
Problem 6.682easy✓ Nihil obstat
Solve \( \displaystyle y' = - 2 x + \cos{\left(x \right)} \) with \( \displaystyle y(0) = -4 \).
Problem 6.684easy✓ Every equation proved
Solve \( \displaystyle y' = 2 x e^{x^{2}} \) with \( \displaystyle y(1) = 5 \).
Problem 6.686easy✓ Nihil obstat
Solve \( \displaystyle y' = \sin{\left(2 x \right)} \) with \( \displaystyle y(0) = 6 \).
Problem 6.687easy✓ Nihil obstat
Solve \( \displaystyle y' = \frac{1}{x} \) with \( \displaystyle y(1) = 4 \).
Problem 6.739easy✓ Every equation proved
Solve \( \displaystyle y' = 2 x e^{x^{2}} \) with \( \displaystyle y(0) = 0 \).
Problem 6.740easy✓ Every equation proved
Solve \( \displaystyle y' = - 2 x + \cos{\left(x \right)} \) with \( \displaystyle y(0) = 8 \).
Problem 6.741easy✓ Every equation proved
Solve \( \displaystyle y' = 2 x e^{x^{2}} \) with \( \displaystyle y(1) = 7 \).
Problem 6.742easy✓ Nihil obstat
Solve \( \displaystyle y' = \sin{\left(2 x \right)} \) with \( \displaystyle y(0) = -1 \).
Problem 6.743easy✓ Every equation proved
Find the general solution of \( \displaystyle y' = \frac{1}{x} \).
Problem 6.744easy✓ Every equation proved
Solve \( \displaystyle y' = 3 x + e^{x} \) with \( \displaystyle y(1) = 7 \).
Problem 6.745easy✓ Nihil obstat
Solve \( \displaystyle y' = 3 x + e^{x} \) with \( \displaystyle y(1) = -1 \).
Problem 6.746easy✓ Every equation proved
Solve \( \displaystyle y' = \sin{\left(2 x \right)} \) with \( \displaystyle y(0) = -3 \).
Problem 6.748easy✓ Nihil obstat
Find the general solution of \( \displaystyle y'' = x^{2} - 1 \).
Problem 6.747medium✓ Nihil obstat