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Differential Equations · First-Order Equations

Linear Equations — Practice Problems and Worked Solutions

Use integrating factors for driven first-order systems.

The formulas you need here

Integrating factor

y^{'} + P (x) y = Q (x), μ = e^{\int P d x}, (μ y)^{'} = μ Q

What it says: Multiply through by μ and the left side collapses into a single product derivative.
When to reach for it: First-order linear equations that will not separate.
Watch out: The equation must be in standard form with the coefficient of y′ equal to 1 before you read off P.

y^{'} + 2 y = e^{x} : μ = e^{2 x} \Rightarrow y = \frac{1}{3} e^{x} + C e^{- 2 x}

The whole method exists to make the left side an exact derivative.

17 worked problems

Every step is generated and checked by a computer algebra system.

1.
Solve the linear initial-value problem.
$y^{'} + 1 y = 2, y (0) = 0$
Answer: $2 - 2 e^{- x}$
Step-by-step solution
1. $μ (x) = e^{1 x}$
2. $y = 2 - 2 e^{- x}$
2.
Solve the linear initial-value problem.
$y^{'} + 2 y = 3, y (0) = 1$
Answer: $\frac{3}{2} - \frac{e ^{- 2 x}}{2}$
Step-by-step solution
1. $μ (x) = e^{2 x}$
2. $y = \frac{3}{2} - \frac{e ^{- 2 x}}{2}$
3.
Solve the linear initial-value problem.
$y^{'} + 3 y = - 1, y (0) = 2$
Answer: $- \frac{1}{3} + \frac{7 e ^{- 3 x}}{3}$
Step-by-step solution
1. $μ (x) = e^{3 x}$
2. $y = - \frac{1}{3} + \frac{7 e ^{- 3 x}}{3}$
4.
Solve the linear initial-value problem.
$y^{'} + - 1 y = 2, y (0) = 1$
Answer: $3 e^{x} - 2$
Step-by-step solution
1. $μ (x) = e^{- 1 x}$
2. $y = 3 e^{x} - 2$
5.
Solve the linear initial-value problem.
$y^{'} + 2 y = - 4, y (0) = - 1$
Answer: $- 2 + e^{- 2 x}$
Step-by-step solution
1. $μ (x) = e^{2 x}$
2. $y = - 2 + e^{- 2 x}$
6.
Solve the linear initial-value problem.
$y^{'} + 4 y = 2, y (0) = 3$
Answer: $\frac{1}{2} + \frac{5 e ^{- 4 x}}{2}$
Step-by-step solution
1. $μ (x) = e^{4 x}$
2. $y = \frac{1}{2} + \frac{5 e ^{- 4 x}}{2}$
7.
Solve the linear initial-value problem.
$y^{'} + 1 y = 4, y (0) = 2$
Answer: $4 - 2 e^{- x}$
Step-by-step solution
1. $μ (x) = e^{1 x}$
2. $y = 4 - 2 e^{- x}$
8.
Solve the linear initial-value problem.
$y^{'} + 2 y = 6, y (0) = 0$
Answer: $3 - 3 e^{- 2 x}$
Step-by-step solution
1. $μ (x) = e^{2 x}$
2. $y = 3 - 3 e^{- 2 x}$
9.
Solve the linear initial-value problem.
$y^{'} + 3 y = 3, y (0) = 1$
Answer: $1$
Step-by-step solution
1. $μ (x) = e^{3 x}$
2. $y = 1$
10.
Solve the linear initial-value problem.
$y^{'} + - 2 y = 4, y (0) = 3$
Answer: $5 e^{2 x} - 2$
Step-by-step solution
1. $μ (x) = e^{- 2 x}$
2. $y = 5 e^{2 x} - 2$
11.
Solve the linear initial-value problem.
$y^{'} + 5 y = 10, y (0) = 1$
Answer: $2 - e^{- 5 x}$
Step-by-step solution
1. $μ (x) = e^{5 x}$
2. $y = 2 - e^{- 5 x}$
12.
Solve the linear initial-value problem.
$y^{'} + 1 y = - 2, y (0) = 4$
Answer: $- 2 + 6 e^{- x}$
Step-by-step solution
1. $μ (x) = e^{1 x}$
2. $y = - 2 + 6 e^{- x}$
13.
Solve the linear initial-value problem.
$y^{'} + 4 y = 8, y (0) = 0$
Answer: $2 - 2 e^{- 4 x}$
Step-by-step solution
1. $μ (x) = e^{4 x}$
2. $y = 2 - 2 e^{- 4 x}$
14.
Solve the linear initial-value problem.
$y^{'} + 2 y = 2, y (0) = 5$
Answer: $1 + 4 e^{- 2 x}$
Step-by-step solution
1. $μ (x) = e^{2 x}$
2. $y = 1 + 4 e^{- 2 x}$
15.
Solve the linear initial-value problem.
$y^{'} + 3 y = 9, y (0) = - 1$
Answer: $3 - 4 e^{- 3 x}$
Step-by-step solution
1. $μ (x) = e^{3 x}$
2. $y = 3 - 4 e^{- 3 x}$
16.
Solve the linear initial-value problem.
$y^{'} + 6 y = 12, y (0) = 2$
Answer: $2$
Step-by-step solution
1. $μ (x) = e^{6 x}$
2. $y = 2$
17.
Solve the linear initial-value problem.
$y^{'} + 1 y = 3, y (0) = - 2$
Answer: $3 - 5 e^{- x}$
Step-by-step solution
1. $μ (x) = e^{1 x}$
2. $y = 3 - 5 e^{- x}$

The formulas you need here

Integrating factor

17 worked problems

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