Ch 05/Linear Differential Equations of First Order/H.K. Dass 3.9

Linear Differential Equations of First Order

A first-order linear DE has the form dy/dx + P(x)y = Q(x). Multiply by the integrating factor e^(integral P dx) and the left side becomes an exact derivative.

IThe Standard Form and I.F. Method

A first-order linear DE:


Step 1: Compute the Integrating Factor: I.F.


Step 2: Solution:


Multiplying by I.F. makes the left side become , which integrates directly.

Note

. This is the most common trick in finding I.F.

Example 1 (Easy)easy

Solve .

Show step-by-step solution+
I.F.
Apply formula
Answer
Example 2 (Medium)medium

Solve .

Show step-by-step solution+
Standard form
.
I.F.
Answer
Example 3 (Linear in x)hard

Solve .

Show step-by-step solution+
Rewrite
. Linear in .
I.F.
Evaluate RHS by substitution
Put . RHS becomes .
Answer
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