First Order Linear Differential Equations: Solving Method

A first-order linear differential equation has the standard form dy/dx + P(x)y = Q(x), where P and Q are functions of x alone and y and its derivative appear only to the first power, never multiplied together or squared. Every such equation is solved by one universal tool: the integrating factor.

This matters because linearity is everywhere in science: electric circuits, mixing tanks, cooling bodies, and investments with deposits all lead to first-order linear equations. Once the equation is in standard form, the solution is pure routine.

Key Concepts

• Standard form: the coefficient of dy/dx must be 1. If the equation is x·dy/dx + 2y = x³, divide everything by x first.
• Linearity test: no y², no y·y′, no sin(y) or ey. P(x) and Q(x) may be any functions of x, including constants.
• Integrating factor: μ(x) = e^(∫P(x) dx). Multiplying through by μ turns the left side into the exact derivative d/dx(μy).
• Solution formula: μy = ∫μQ dx + C, then divide by μ to isolate y.
• Constant of integration: add C when integrating μQ; dividing by μ at the end distributes over it correctly.

Formulas

• Standard form: dy/dx + P(x)y = Q(x).
• Integrating factor: μ(x) = e^(∫P(x) dx) (any antiderivative works; the constant merges into C).
• Solution: y = (1/μ)(∫μQ dx + C).

Worked Example

Solve dy/dx + (1/x)y = x² for x > 0.

1. Confirm standard form: the coefficient of dy/dx is 1, with P(x) = 1/x and Q(x) = x².
2. Find the integrating factor: ∫P(x) dx = ∫(1/x) dx = ln x, so μ = e^(ln x) = x.
3. Multiply through: x·dy/dx + y = x³. The left side is exactly d/dx(xy) by the product rule.
4. Integrate: d/dx(xy) = x³ gives xy = ∫x³ dx = x⁴/4 + C.
5. Solve for y: y = x³/4 + C/x.
6. Verify: y′ = 3x²/4 − C/x². Then y′ + y/x = 3x²/4 − C/x² + x²/4 + C/x² = x². Correct.

Common Misconceptions

• Forgetting to divide into standard form. Reading P(x) off an equation like 2y′ + 4xy = 6 without dividing by 2 gives the wrong integrating factor.
• Sign errors in μ. The exponent is +∫P dx exactly as P appears in dy/dx + P(x)y = Q(x); a minus sign in P stays a minus sign.
• Forgetting the final division. Integrating gives μy, not y; the last step of dividing by μ is part of the method, not optional.

Real-World Applications

• Electric circuits: an RL circuit obeys L·dI/dt + RI = V, a first-order linear equation for the current.
• Mixing tanks: salt concentration in a tank with inflow and outflow follows a linear equation.
• Cooling with a heat source: temperature with both cooling and internal heating is linear.
• Finance: an investment earning interest while receiving regular deposits satisfies a linear equation.

First-order linear equations are solved by a fixed pipeline: divide into standard form, compute μ = e^(∫P dx), recognise (μy)′, integrate, and divide by μ. It never varies, so accuracy comes entirely from careful algebra at each step.