ORDINARY DIFFERENTIAL EQUATIONS: LINEAR COEFFICIENTS
Ordinary Differential Equations
with Linear Coefficients
Comprehensive Theoretical Manual, Derivation Framework,
and Complete Solution Compendium
Course Unit: Applied Mathematics / First-Order Differential Equations
Topic: Non-Homogeneous Linear Coefficient Differential Equations
Edition: Monochromatic Academic Study Edition (2026)
Abstract
This course manual presents an exhaustive, rigorous, and completely original treat-
ment of first-order ordinary differential equations featuring linear coefficients of the form
(a1 x + b1 y + c1 )dx + (a2 x + b2 y + c2 )dy = 0. Grounded in fundamental geometric prin-
ciples and coordinate transformations, this manual covers the complete theoretical tax-
onomy—distinguishing between intersecting coefficient lines (Case 1) and parallel coeffi-
cient lines (Case 2). Detailed mathematical derivations demonstrate the origin translation
method, constant elimination theorems, differential transformations, variable separation
techniques, and double back-substitution algorithms. Complete, line-by-line worked solu-
tions are provided for every problem in the course reference set, accompanied by diagnostic
decision frameworks, common exam traps, and active recall self-assessment instruments.
1
,ORDINARY DIFFERENTIAL EQUATIONS: LINEAR COEFFICIENTS CONTENTS
Contents
1 Geometric Foundations & Taxonomy Framework 5
1.1 Standard Formulation and Definition . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 The System Determinant and Classification Theorem . . . . . . . . . . . . . . . . 5
1.3 Comparative Methodological Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Theoretical Derivations & Methodological Proofs 7
2.1 Proof of Constant Elimination under Axis Translation . . . . . . . . . . . . . . . 7
2.2 Homogeneous Reduction Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Option A: Setting u = vz . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Option B: Setting v = uz . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Analytical Integration Taxonomy for z-Integrands . . . . . . . . . . . . . . . . . 8
3 Exhaustive Step-by-Step Problem Solutions 9
3.1 Problem 1: Solution for (2x − y − 4)dx + (3x + 2y + 1)dy = 0 . . . . . . . . . . . 9
3.1.1 Step 1: Calculate Intersection Point (h, k) . . . . . . . . . . . . . . . . . . 9
3.1.2 Step 2: Coordinate Translation . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.3 Step 3: Homogeneous Substitution . . . . . . . . . . . . . . . . . . . . . . 9
3.1.4 Step 4: Variable Separation and Integration . . . . . . . . . . . . . . . . . 10
3.1.5 Step 5: Double Back-Substitution . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Problem 2: Solution for (2x + 3y − 1)dx + (3x + y + 2)dy = 0 . . . . . . . . . . . 12
3.2.1 Step 1: Calculate Intersection Point (h, k) . . . . . . . . . . . . . . . . . . 12
3.2.2 Step 2: Coordinate Translation . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.3 Step 3: Homogeneous Substitution . . . . . . . . . . . . . . . . . . . . . . 12
3.2.4 Step 4: Variable Separation and Integration . . . . . . . . . . . . . . . . . 12
3.2.5 Step 5: Double Back-Substitution . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Problem 3: Solution for (3x + 2y + 2)dx + (x + 4)dy = 0 . . . . . . . . . . . . . . 14
3.3.1 Step 1: Calculate Intersection Point (h, k) . . . . . . . . . . . . . . . . . . 14
3.3.2 Step 2: Coordinate Translation . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3.3 Step 3: Homogeneous Substitution . . . . . . . . . . . . . . . . . . . . . . 14
3.3.4 Step 4: Variable Separation and Partial Fractions . . . . . . . . . . . . . . 14
3.3.5 Step 5: Double Back-Substitution . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Problem 4: Solution for (y + 3)dx − (x − y − 5)dy = 0 . . . . . . . . . . . . . . . 16
3.4.1 Step 1: Calculate Intersection Point (h, k) . . . . . . . . . . . . . . . . . . 16
3.4.2 Step 2: Coordinate Translation . . . . . . . . . . . . . . . . . . . . . . . . 16
2
, ORDINARY DIFFERENTIAL EQUATIONS: LINEAR COEFFICIENTS CONTENTS
3.4.3 Step 3: Analytical Resolution via Exact Differential . . . . . . . . . . . . 16
3.4.4 Step 4: Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4.5 Step 5: Double Back-Substitution . . . . . . . . . . . . . . . . . . . . . . 16
3.5 Problem 5: Comprehensive Solution for Lecture Demonstration . . . . . . . . . . 17
3.5.1 Step 1: Calculate Intersection Point (h, k) . . . . . . . . . . . . . . . . . . 17
3.5.2 Step 2: Coordinate Translation . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5.3 Step 3: Substitution v = uz (Lecture Method) . . . . . . . . . . . . . . . 17
3.5.4 Step 4: Variable Separation and Partial Fraction Decomposition . . . . . 17
3.5.5 Step 5: Double Back-Substitution . . . . . . . . . . . . . . . . . . . . . . 18
3.6 Problem 6: Solution for (5x − y − 9)dx + (3x + 2y − 8)dy = 0 . . . . . . . . . . . 19
3.6.1 Step 1: Calculate Intersection Point (h, k) . . . . . . . . . . . . . . . . . . 19
3.6.2 Step 2: Coordinate Translation . . . . . . . . . . . . . . . . . . . . . . . . 19
3.6.3 Step 3: Homogeneous Substitution . . . . . . . . . . . . . . . . . . . . . . 19
3.6.4 Step 4: Variable Separation and Integration . . . . . . . . . . . . . . . . . 19
3.6.5 Step 5: Double Back-Substitution . . . . . . . . . . . . . . . . . . . . . . 19
3.7 Problem 7: Solution for (4x − y + 7)dx − (x + 2y − 5)dy = 0 . . . . . . . . . . . 20
3.7.1 Step 1: Calculate Intersection Point (h, k) . . . . . . . . . . . . . . . . . . 20
3.7.2 Step 2: Coordinate Translation . . . . . . . . . . . . . . . . . . . . . . . . 20
3.7.3 Step 3: Homogeneous Substitution . . . . . . . . . . . . . . . . . . . . . . 20
3.7.4 Step 4: Variable Separation and Integration . . . . . . . . . . . . . . . . . 20
3.7.5 Step 5: Double Back-Substitution . . . . . . . . . . . . . . . . . . . . . . 20
3.8 Parallel Case Study: Solution for (2x + 3y + 1)dx + (4x + 6y − 5)dy = 0 . . . . . 21
3.8.1 Step 1: Check System Determinant . . . . . . . . . . . . . . . . . . . . . . 21
3.8.2 Step 2: Composite Substitution . . . . . . . . . . . . . . . . . . . . . . . . 21
3.8.3 Step 3: Substitution and Separation . . . . . . . . . . . . . . . . . . . . . 21
3.8.4 Step 4: Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.8.5 Step 5: Back-Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Diagnostic Flowchart & Decision Trees 22
4.1 Systematic Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Exam Pitfalls & Professor Traps 23
6 10-Question High-Yield Practice Problem Set 24
7 Active Recall Self-Test & Inverted Answer Key 25
7.1 Conceptual Recall Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3
Ordinary Differential Equations
with Linear Coefficients
Comprehensive Theoretical Manual, Derivation Framework,
and Complete Solution Compendium
Course Unit: Applied Mathematics / First-Order Differential Equations
Topic: Non-Homogeneous Linear Coefficient Differential Equations
Edition: Monochromatic Academic Study Edition (2026)
Abstract
This course manual presents an exhaustive, rigorous, and completely original treat-
ment of first-order ordinary differential equations featuring linear coefficients of the form
(a1 x + b1 y + c1 )dx + (a2 x + b2 y + c2 )dy = 0. Grounded in fundamental geometric prin-
ciples and coordinate transformations, this manual covers the complete theoretical tax-
onomy—distinguishing between intersecting coefficient lines (Case 1) and parallel coeffi-
cient lines (Case 2). Detailed mathematical derivations demonstrate the origin translation
method, constant elimination theorems, differential transformations, variable separation
techniques, and double back-substitution algorithms. Complete, line-by-line worked solu-
tions are provided for every problem in the course reference set, accompanied by diagnostic
decision frameworks, common exam traps, and active recall self-assessment instruments.
1
,ORDINARY DIFFERENTIAL EQUATIONS: LINEAR COEFFICIENTS CONTENTS
Contents
1 Geometric Foundations & Taxonomy Framework 5
1.1 Standard Formulation and Definition . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 The System Determinant and Classification Theorem . . . . . . . . . . . . . . . . 5
1.3 Comparative Methodological Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Theoretical Derivations & Methodological Proofs 7
2.1 Proof of Constant Elimination under Axis Translation . . . . . . . . . . . . . . . 7
2.2 Homogeneous Reduction Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Option A: Setting u = vz . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Option B: Setting v = uz . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Analytical Integration Taxonomy for z-Integrands . . . . . . . . . . . . . . . . . 8
3 Exhaustive Step-by-Step Problem Solutions 9
3.1 Problem 1: Solution for (2x − y − 4)dx + (3x + 2y + 1)dy = 0 . . . . . . . . . . . 9
3.1.1 Step 1: Calculate Intersection Point (h, k) . . . . . . . . . . . . . . . . . . 9
3.1.2 Step 2: Coordinate Translation . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1.3 Step 3: Homogeneous Substitution . . . . . . . . . . . . . . . . . . . . . . 9
3.1.4 Step 4: Variable Separation and Integration . . . . . . . . . . . . . . . . . 10
3.1.5 Step 5: Double Back-Substitution . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Problem 2: Solution for (2x + 3y − 1)dx + (3x + y + 2)dy = 0 . . . . . . . . . . . 12
3.2.1 Step 1: Calculate Intersection Point (h, k) . . . . . . . . . . . . . . . . . . 12
3.2.2 Step 2: Coordinate Translation . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.3 Step 3: Homogeneous Substitution . . . . . . . . . . . . . . . . . . . . . . 12
3.2.4 Step 4: Variable Separation and Integration . . . . . . . . . . . . . . . . . 12
3.2.5 Step 5: Double Back-Substitution . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Problem 3: Solution for (3x + 2y + 2)dx + (x + 4)dy = 0 . . . . . . . . . . . . . . 14
3.3.1 Step 1: Calculate Intersection Point (h, k) . . . . . . . . . . . . . . . . . . 14
3.3.2 Step 2: Coordinate Translation . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3.3 Step 3: Homogeneous Substitution . . . . . . . . . . . . . . . . . . . . . . 14
3.3.4 Step 4: Variable Separation and Partial Fractions . . . . . . . . . . . . . . 14
3.3.5 Step 5: Double Back-Substitution . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Problem 4: Solution for (y + 3)dx − (x − y − 5)dy = 0 . . . . . . . . . . . . . . . 16
3.4.1 Step 1: Calculate Intersection Point (h, k) . . . . . . . . . . . . . . . . . . 16
3.4.2 Step 2: Coordinate Translation . . . . . . . . . . . . . . . . . . . . . . . . 16
2
, ORDINARY DIFFERENTIAL EQUATIONS: LINEAR COEFFICIENTS CONTENTS
3.4.3 Step 3: Analytical Resolution via Exact Differential . . . . . . . . . . . . 16
3.4.4 Step 4: Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4.5 Step 5: Double Back-Substitution . . . . . . . . . . . . . . . . . . . . . . 16
3.5 Problem 5: Comprehensive Solution for Lecture Demonstration . . . . . . . . . . 17
3.5.1 Step 1: Calculate Intersection Point (h, k) . . . . . . . . . . . . . . . . . . 17
3.5.2 Step 2: Coordinate Translation . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5.3 Step 3: Substitution v = uz (Lecture Method) . . . . . . . . . . . . . . . 17
3.5.4 Step 4: Variable Separation and Partial Fraction Decomposition . . . . . 17
3.5.5 Step 5: Double Back-Substitution . . . . . . . . . . . . . . . . . . . . . . 18
3.6 Problem 6: Solution for (5x − y − 9)dx + (3x + 2y − 8)dy = 0 . . . . . . . . . . . 19
3.6.1 Step 1: Calculate Intersection Point (h, k) . . . . . . . . . . . . . . . . . . 19
3.6.2 Step 2: Coordinate Translation . . . . . . . . . . . . . . . . . . . . . . . . 19
3.6.3 Step 3: Homogeneous Substitution . . . . . . . . . . . . . . . . . . . . . . 19
3.6.4 Step 4: Variable Separation and Integration . . . . . . . . . . . . . . . . . 19
3.6.5 Step 5: Double Back-Substitution . . . . . . . . . . . . . . . . . . . . . . 19
3.7 Problem 7: Solution for (4x − y + 7)dx − (x + 2y − 5)dy = 0 . . . . . . . . . . . 20
3.7.1 Step 1: Calculate Intersection Point (h, k) . . . . . . . . . . . . . . . . . . 20
3.7.2 Step 2: Coordinate Translation . . . . . . . . . . . . . . . . . . . . . . . . 20
3.7.3 Step 3: Homogeneous Substitution . . . . . . . . . . . . . . . . . . . . . . 20
3.7.4 Step 4: Variable Separation and Integration . . . . . . . . . . . . . . . . . 20
3.7.5 Step 5: Double Back-Substitution . . . . . . . . . . . . . . . . . . . . . . 20
3.8 Parallel Case Study: Solution for (2x + 3y + 1)dx + (4x + 6y − 5)dy = 0 . . . . . 21
3.8.1 Step 1: Check System Determinant . . . . . . . . . . . . . . . . . . . . . . 21
3.8.2 Step 2: Composite Substitution . . . . . . . . . . . . . . . . . . . . . . . . 21
3.8.3 Step 3: Substitution and Separation . . . . . . . . . . . . . . . . . . . . . 21
3.8.4 Step 4: Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.8.5 Step 5: Back-Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Diagnostic Flowchart & Decision Trees 22
4.1 Systematic Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Exam Pitfalls & Professor Traps 23
6 10-Question High-Yield Practice Problem Set 24
7 Active Recall Self-Test & Inverted Answer Key 25
7.1 Conceptual Recall Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3
