Forms of Solving Differential Equations>Linear Differential Equations

A first-order and first-degree differential equation of the form

\[ \boxed{\,\frac{dy}{dx} + Py = Q\,} \]

where \[ P \] and \[ Q \] are constants or functions of \[ x \] only, is called a first-order linear differential equation.

Similarly, if \[ x \] is treated as a function of \[ y, \] it may be written as

\[ \boxed{\,\frac{dx}{dy} + P_{1}x = Q_{1}\,} \]

where \[ P_{1} \] and \[ Q_{1} \] are constants or functions of \[ y \] only.

• Write the given differential equation in standard linear form.

• Identify P and Q(or P₁ and Q₁).

• Find the integrating factor:

  \[ \text{For } \frac{dy}{dx} + Py = Q, \quad \text{I.F.} = e^{\int P\,dx} \]

  \[ \text{For } \frac{dx}{dy} + P_{1}x = Q_{1}, \quad \text{I.F.} = e^{\int P_{1}\,dy} \]

• Multiply the complete equation by the integrating factor.

• Convert the left-hand side into the derivative of a product.

• Integrate both sides.

• Apply the initial condition, if given, and write the final answer clearly.

Solve \[\frac{dy}{dx} - y = \cos x\]

Given equation:

This is already in standard form with P = -1 and Q = cos x.

Integrating factor:

On multiplying throughout by \[e^{-x}\], the equation becomes:

1. Write the equation in the form dy/dx + Py = Q
2. Identify P and Q or P₁ and Q₁
3. Find I.F. = 

   \[ \frac{dy}{dx} + Py = Q \Rightarrow \text{I.F.} = e^{\int P\,dx} \]

   \[ \frac{dx}{dy} + P_{1}x = Q_{1} \Rightarrow \text{I.F.} = e^{\int P_{1}\,dy} \]

4. Multiply the whole equation by I.F.
5. Integrate and get a solution.