Forms of Solving Differential Equations>Linear Differential Equations

A linear differential equation of first order and first degree is
\[\frac{\mathrm{d}y}{\mathrm{d}x}+\mathrm{P}y=\mathrm{Q}\], where P and Q are the functions of x or constants. Its general solution is  \[y.\left(\mathrm{I.F.}\right)=\int\mathrm{Q.}\left(\mathrm{I.F.}\right)\mathrm{d}x+\mathrm{c}\] and the function \[\mathrm{e}^{\int\mathrm{Pdx}}\] is called the integrating factor (I.F.) of the given equation.

• Write the given differential equation in standard linear form.

• Identify P and Q.

• Find the integrating factor using \[e^{\int P \, dx}\] or \[e^{\int P \, dy}\].

• 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
3. Find I.F. = \[\mathrm{e}^{\int\mathrm{Pdx}}\]
4. Multiply the whole equation by I.F.
5. Integrate and get a solution.