Linear Differential Equations of First Order
An equation of the form dy/dx + P(x) y = Q(x) is solved by multiplying by the integrating factor exp(integral of P dx).
What is Linear Differential Equations of First Order?
An equation of the form dy/dx + P(x) y = Q(x) is solved by multiplying by the integrating factor exp(integral of P dx).
Key formula / rule: Standard form must have coefficient 1 on dy/dx before reading P and Q.
Key points
- Derive the integrating factor for a first-order linear equation.
- Compute the general solution of a first-order linear equation.
Common exam trap
Reading P from an equation not in standard form.
Definitions
- Term
Linear Differential Equations of First Order
- Meaning
An equation of the form dy/dx + P(x) y = Q(x) is solved by multiplying by the integrating factor exp(integral of P dx).
- Term
Linear Differential Equations of First Order — explanation
- Meaning
The integrating factor turns the left side into an exact derivative of a product, so a single integration finishes the problem. Writing the equation in standard form first is essential.
Learning objectives
Derive the integrating factor for a first-order linear equation.
Compute the general solution of a first-order linear equation.
Formulae
- Key point
Standard form must have coefficient 1 on dy/dx before reading P and Q.
- Key point
The integrating factor is exp(integral P dx) and needs no constant.
- Key point
After multiplying, the left side is d/dx of (IF × y).
Prerequisites
MAT-U2-DE-T1-S1-C2
Common mistakes
Reading P from an equation not in standard form.
Including a constant of integration inside the integrating factor.
Keywords
Linear
Differential
Equations
First
Practice preview
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Identify P(x) and Q(x) for the differential equation dy/dx + (2x/(1+x^2))y = sin(x).…
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Find the general solution of the differential equation (1 + x^2) dy/dx + 2xy = cos(x).…
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