APM2611 Study Notes
Applied Differential Equations – Formula Sheets, Methods & Worked Examples
Based on Zill, Differential Equations with Boundary-Value Problems, 8th Ed.
Covering module outcomes 2.2.1 – 2.2.6
How to use this guide. Each section below gives you: a boxed formula sheet with everything you need
to memorise, a step-by-step method for solving that type of problem, and worked examples (including
trickier variations) that follow the steps exactly. Work through the examples with a pen before checking the
solution.
Contents
PART I – Chapter 1: Introduction to Differential Equations 2
1.1 – 1.2 Definitions, Terminology & Initial-Value Problems 2
1.3 Differential Equations as Mathematical Models 4
PART II – Chapter 2: First-Order Differential Equations 5
2.1 Solution Curves Without a Solution 5
2.2 Separable Equations 6
2.3 Linear Equations 7
2.4 Exact Equations 8
2.5 Solutions by Substitution 9
2.6 A Numerical Method – Euler’s Method 10
PART III – Chapter 3: Modeling with First-Order Differential Equations 12
3.1 Linear Models 12
3.2 Nonlinear Models 13
3.3 Modeling with Systems of First-Order DEs 14
PART IV – Chapter 4: Higher-Order Differential Equations 15
4.1 Preliminary Theory – Linear Equations 15
4.2 Reduction of Order 15
4.3 Homogeneous Linear Equations with Constant Coefficients 16
4.4 – 4.5 Undetermined Coefficients (Superposition & Annihilator) 17
4.6 Variation of Parameters 18
4.7 Cauchy–Euler Equation 19
4.8 Green’s Functions (Overview) 20
4.9 Solving Systems of Linear DEs by Elimination 21
4.10 Nonlinear Differential Equations (Overview) 21
,APM2611 – Applied Differential Equations CONTENTS
PART V – Chapter 6: Series Solutions of Linear Equations 23
6.1 Review of Power Series 23
6.2 Solutions About Ordinary Points 24
6.3 Solutions About Singular Points – Frobenius Method 25
6.4 Special Functions – Bessel and Legendre 26
PART VI – Chapter 7: The Laplace Transform 28
7.1 Definition of the Laplace Transform 28
7.2 Inverse Transforms & Transforms of Derivatives 29
7.3 Operational Properties I – Translation Theorems 30
7.4 Operational Properties II 31
7.5 The Dirac Delta Function 32
7.6 Systems of Linear Differential Equations 33
PART VII – Chapter 11: Fourier Series 34
11.1 Orthogonal Functions 34
11.2 Fourier Series 34
11.3 Fourier Cosine and Sine Series 35
11.4 Sturm–Liouville Problem 36
11.5 Bessel and Legendre Series 37
PART VIII – Chapter 12: Boundary-Value Problems in Rectangular Coordinates 38
12.1 Separable Partial Differential Equations 38
12.2 Classical PDEs and Boundary-Value Problems 38
12.3 Heat Equation 39
12.4 Wave Equation 40
12.5 Laplace’s Equation 41
12.6 Nonhomogeneous Boundary-Value Problems 42
12.7 Orthogonal Series Expansions 43
12.8 Higher-Dimensional Problems 43
2
,APM2611 – Applied Differential Equations Chapter 1: Introduction
PART I – Chapter 1: Introduction to Differential Equations
1.1 – 1.2 Definitions, Terminology & Initial-Value Problems
FORMULA SHEET
Differential equation (DE): an equation containing derivatives of one or more unknown functions.
dy
ODE (ordinary): unknown function depends on ONE variable, e.g. + 5y = ex .
dx
∂2u ∂u
PDE (partial): unknown function depends on TWO or more variables, e.g. = .
∂x2 ∂t
Order: the order of the highest derivative present.
Degree: power of the highest-order derivative (once the equation is polynomial in derivatives).
Linear DE of order n: can be written
dn y dn−1 y dy
an (x) + an−1 (x) + · · · + a1 (x) + a0 (x)y = g(x)
dxn dxn−1 dx
i.e. y and all its derivatives appear to the power 1, and no products such as yy ′ or sin y.
Nonlinear: any DE that fails the linearity test above (e.g. y ′′ + sin y = 0, yy ′ = x).
Solution: a function ϕ(x) that, substituted into the DE, reduces it to an identity on some interval I
(interval of definition).
Explicit solution: y = ϕ(x). Implicit solution: G(x, y) = 0.
Family of solutions: contains arbitrary constants c1 , c2 , . . . A particular solution has specific values for
the constants.
Initial-Value Problem (IVP): the DE together with initial condition(s) at one point x0 : y(x0 ) = y0 ,
y ′ (x0 ) = y1 , . . .
METHOD – STEP BY STEP Classify a DE (Outcome 2.2.1)
1. Count the number of independent variables → ODE (one) or PDE (more than one).
2. Identify the highest derivative → that is the order.
3. Check: is the unknown function and every one of its derivatives raised only to the power 1, with coefficients
depending on the independent variable only (not on y)? If yes → linear; otherwise → nonlinear.
METHOD – STEP BY STEP Verify that a given function is a solution
1. Differentiate the proposed solution y = ϕ(x) as many times as the order of the DE requires.
2. Substitute ϕ(x) and its derivatives into the DE.
3. Simplify. If the left side equals the right side for all x on the stated interval → it is a solution.
4. If the solution is given implicitly, differentiate the relation implicitly and show the DE is satisfied.
EXAMPLE Classification
Classify each DE: order, linear/nonlinear, ODE/PDE.
d2 y dy
(a) −2 +y =0
dx2 dx
(b) (1 − y)y ′ + 2y = ex
∂2u ∂2u
(c) + 2 =0
∂x2 ∂y
3
, APM2611 – Applied Differential Equations Chapter 1: Introduction
2
d3 y
dy
(d) +x =0
dx3 dx
Solution.
(a) Highest derivative is y ′′ ⇒ order 2, ODE. Coefficients of y ′′ , y ′ , y are constants (do not depend on y) and
each term is degree 1 in y and its derivatives ⇒ linear.
(b) Highest derivative is y ′ ⇒ order 1, ODE. The coefficient of y ′ is (1 − y), which depends on y ⇒ nonlinear.
(c) Two independent variables x, y with partial derivatives ⇒ PDE, order 2, linear.
(d) Highest derivative is y ′′′ , but it is squared ⇒ order 3, ODE, nonlinear (a derivative is raised to a power
other than 1).
EXAMPLE Verifying a solution – straightforward
Show that y = c1 e2x + c2 e−2x is a solution of y ′′ − 4y = 0 for any constants c1 , c2 .
Solution. y ′ = 2c1 e2x − 2c2 e−2x , y ′′ = 4c1 e2x + 4c2 e−2x . Then
y ′′ − 4y = (4c1 e2x + 4c2 e−2x ) − 4(c1 e2x + c2 e−2x ) = 0.✓
EXAMPLE Verifying an implicit / tricky solution
dy √
Show that x2 + y 2 − 4 = 0 is an implicit solution of y = −x on −2 < x < 2, and explain why y = 4 − x2
dx
alone is NOT the complete picture.
Solution. Differentiate x2 + y 2 − 4 = 0 implicitly with respect to x:
dy dy
2x + 2y = 0 =⇒ y = −x.✓
dx dx
√ √
This works for both branches y = 4 − x2 (upper semicircle) and y = − 4 − x2 (lower semicircle) – each is
a valid explicit solution on the open interval (−2, 2) (endpoints excluded because y ′ is undefined there, as the
tangent is vertical). This shows an implicit solution can represent more than one explicit solution – a common
trick question.
EXAMPLE IVP – solve for the arbitrary constant
The one-parameter family y = ce−x solves y ′ + y = 0. Solve the IVP y ′ + y = 0, y(0) = 3.
Solution. Apply the step-by-step method:
1. General solution: y = ce−x .
2. Substitute the initial condition x = 0, y = 3: 3 = ce0 = c ⇒ c = 3.
3. Particular solution: y = 3e−x .
COMMON PITFALLS
A coefficient multiplying y or a derivative that itself depends on y (like sin y, y 2 , yy ′ ) instantly makes the
DE nonlinear – this is the most commonly tested trick.
Don’t confuse order (highest derivative) with degree (power of that highest derivative).
An nth-order IVP needs exactly n initial conditions, all specified at the same point x0 ; if conditions are
given at different points, it is a Boundary-Value Problem (BVP) instead.
1.3 Differential Equations as Mathematical Models
4
Applied Differential Equations – Formula Sheets, Methods & Worked Examples
Based on Zill, Differential Equations with Boundary-Value Problems, 8th Ed.
Covering module outcomes 2.2.1 – 2.2.6
How to use this guide. Each section below gives you: a boxed formula sheet with everything you need
to memorise, a step-by-step method for solving that type of problem, and worked examples (including
trickier variations) that follow the steps exactly. Work through the examples with a pen before checking the
solution.
Contents
PART I – Chapter 1: Introduction to Differential Equations 2
1.1 – 1.2 Definitions, Terminology & Initial-Value Problems 2
1.3 Differential Equations as Mathematical Models 4
PART II – Chapter 2: First-Order Differential Equations 5
2.1 Solution Curves Without a Solution 5
2.2 Separable Equations 6
2.3 Linear Equations 7
2.4 Exact Equations 8
2.5 Solutions by Substitution 9
2.6 A Numerical Method – Euler’s Method 10
PART III – Chapter 3: Modeling with First-Order Differential Equations 12
3.1 Linear Models 12
3.2 Nonlinear Models 13
3.3 Modeling with Systems of First-Order DEs 14
PART IV – Chapter 4: Higher-Order Differential Equations 15
4.1 Preliminary Theory – Linear Equations 15
4.2 Reduction of Order 15
4.3 Homogeneous Linear Equations with Constant Coefficients 16
4.4 – 4.5 Undetermined Coefficients (Superposition & Annihilator) 17
4.6 Variation of Parameters 18
4.7 Cauchy–Euler Equation 19
4.8 Green’s Functions (Overview) 20
4.9 Solving Systems of Linear DEs by Elimination 21
4.10 Nonlinear Differential Equations (Overview) 21
,APM2611 – Applied Differential Equations CONTENTS
PART V – Chapter 6: Series Solutions of Linear Equations 23
6.1 Review of Power Series 23
6.2 Solutions About Ordinary Points 24
6.3 Solutions About Singular Points – Frobenius Method 25
6.4 Special Functions – Bessel and Legendre 26
PART VI – Chapter 7: The Laplace Transform 28
7.1 Definition of the Laplace Transform 28
7.2 Inverse Transforms & Transforms of Derivatives 29
7.3 Operational Properties I – Translation Theorems 30
7.4 Operational Properties II 31
7.5 The Dirac Delta Function 32
7.6 Systems of Linear Differential Equations 33
PART VII – Chapter 11: Fourier Series 34
11.1 Orthogonal Functions 34
11.2 Fourier Series 34
11.3 Fourier Cosine and Sine Series 35
11.4 Sturm–Liouville Problem 36
11.5 Bessel and Legendre Series 37
PART VIII – Chapter 12: Boundary-Value Problems in Rectangular Coordinates 38
12.1 Separable Partial Differential Equations 38
12.2 Classical PDEs and Boundary-Value Problems 38
12.3 Heat Equation 39
12.4 Wave Equation 40
12.5 Laplace’s Equation 41
12.6 Nonhomogeneous Boundary-Value Problems 42
12.7 Orthogonal Series Expansions 43
12.8 Higher-Dimensional Problems 43
2
,APM2611 – Applied Differential Equations Chapter 1: Introduction
PART I – Chapter 1: Introduction to Differential Equations
1.1 – 1.2 Definitions, Terminology & Initial-Value Problems
FORMULA SHEET
Differential equation (DE): an equation containing derivatives of one or more unknown functions.
dy
ODE (ordinary): unknown function depends on ONE variable, e.g. + 5y = ex .
dx
∂2u ∂u
PDE (partial): unknown function depends on TWO or more variables, e.g. = .
∂x2 ∂t
Order: the order of the highest derivative present.
Degree: power of the highest-order derivative (once the equation is polynomial in derivatives).
Linear DE of order n: can be written
dn y dn−1 y dy
an (x) + an−1 (x) + · · · + a1 (x) + a0 (x)y = g(x)
dxn dxn−1 dx
i.e. y and all its derivatives appear to the power 1, and no products such as yy ′ or sin y.
Nonlinear: any DE that fails the linearity test above (e.g. y ′′ + sin y = 0, yy ′ = x).
Solution: a function ϕ(x) that, substituted into the DE, reduces it to an identity on some interval I
(interval of definition).
Explicit solution: y = ϕ(x). Implicit solution: G(x, y) = 0.
Family of solutions: contains arbitrary constants c1 , c2 , . . . A particular solution has specific values for
the constants.
Initial-Value Problem (IVP): the DE together with initial condition(s) at one point x0 : y(x0 ) = y0 ,
y ′ (x0 ) = y1 , . . .
METHOD – STEP BY STEP Classify a DE (Outcome 2.2.1)
1. Count the number of independent variables → ODE (one) or PDE (more than one).
2. Identify the highest derivative → that is the order.
3. Check: is the unknown function and every one of its derivatives raised only to the power 1, with coefficients
depending on the independent variable only (not on y)? If yes → linear; otherwise → nonlinear.
METHOD – STEP BY STEP Verify that a given function is a solution
1. Differentiate the proposed solution y = ϕ(x) as many times as the order of the DE requires.
2. Substitute ϕ(x) and its derivatives into the DE.
3. Simplify. If the left side equals the right side for all x on the stated interval → it is a solution.
4. If the solution is given implicitly, differentiate the relation implicitly and show the DE is satisfied.
EXAMPLE Classification
Classify each DE: order, linear/nonlinear, ODE/PDE.
d2 y dy
(a) −2 +y =0
dx2 dx
(b) (1 − y)y ′ + 2y = ex
∂2u ∂2u
(c) + 2 =0
∂x2 ∂y
3
, APM2611 – Applied Differential Equations Chapter 1: Introduction
2
d3 y
dy
(d) +x =0
dx3 dx
Solution.
(a) Highest derivative is y ′′ ⇒ order 2, ODE. Coefficients of y ′′ , y ′ , y are constants (do not depend on y) and
each term is degree 1 in y and its derivatives ⇒ linear.
(b) Highest derivative is y ′ ⇒ order 1, ODE. The coefficient of y ′ is (1 − y), which depends on y ⇒ nonlinear.
(c) Two independent variables x, y with partial derivatives ⇒ PDE, order 2, linear.
(d) Highest derivative is y ′′′ , but it is squared ⇒ order 3, ODE, nonlinear (a derivative is raised to a power
other than 1).
EXAMPLE Verifying a solution – straightforward
Show that y = c1 e2x + c2 e−2x is a solution of y ′′ − 4y = 0 for any constants c1 , c2 .
Solution. y ′ = 2c1 e2x − 2c2 e−2x , y ′′ = 4c1 e2x + 4c2 e−2x . Then
y ′′ − 4y = (4c1 e2x + 4c2 e−2x ) − 4(c1 e2x + c2 e−2x ) = 0.✓
EXAMPLE Verifying an implicit / tricky solution
dy √
Show that x2 + y 2 − 4 = 0 is an implicit solution of y = −x on −2 < x < 2, and explain why y = 4 − x2
dx
alone is NOT the complete picture.
Solution. Differentiate x2 + y 2 − 4 = 0 implicitly with respect to x:
dy dy
2x + 2y = 0 =⇒ y = −x.✓
dx dx
√ √
This works for both branches y = 4 − x2 (upper semicircle) and y = − 4 − x2 (lower semicircle) – each is
a valid explicit solution on the open interval (−2, 2) (endpoints excluded because y ′ is undefined there, as the
tangent is vertical). This shows an implicit solution can represent more than one explicit solution – a common
trick question.
EXAMPLE IVP – solve for the arbitrary constant
The one-parameter family y = ce−x solves y ′ + y = 0. Solve the IVP y ′ + y = 0, y(0) = 3.
Solution. Apply the step-by-step method:
1. General solution: y = ce−x .
2. Substitute the initial condition x = 0, y = 3: 3 = ce0 = c ⇒ c = 3.
3. Particular solution: y = 3e−x .
COMMON PITFALLS
A coefficient multiplying y or a derivative that itself depends on y (like sin y, y 2 , yy ′ ) instantly makes the
DE nonlinear – this is the most commonly tested trick.
Don’t confuse order (highest derivative) with degree (power of that highest derivative).
An nth-order IVP needs exactly n initial conditions, all specified at the same point x0 ; if conditions are
given at different points, it is a Boundary-Value Problem (BVP) instead.
1.3 Differential Equations as Mathematical Models
4
