Linear Equations of Order One
Standard form of a Linear Equation of Order One
𝒊. Linear in 𝒚
𝑑𝑦
𝑑𝑥
+ 𝑃(𝑥)𝑦 = 𝑄(𝑥)
𝑦′ + 𝑃(𝑥)𝑦 = 𝑄(𝑥)
or
where 𝑃(𝑥) and 𝑄(𝑥) are either constants or functions of 𝑥 alone.
𝒊𝒊. Linear in 𝒙
𝑑𝑥
𝑑𝑦
+ 𝑃(𝑦)𝑥 = 𝑄(𝑦)
𝑥′ + 𝑃(𝑦)𝑥 = 𝑄(𝑦)
or
where 𝑃(𝑦) and 𝑄(𝑦) are either constants or functions of 𝑦 alone.
Remark: The equation is linear with respect to a particular variable if that
variable is of the first degree and there is no product of that variable and
its derivative or differential in the equation.
Integrating Factor
A function 𝜑 is an integrating factor of the different equation
𝑀𝑑𝑥 + 𝑁𝑑𝑦 = 0
If 𝝋𝑴𝒅𝒙 + 𝝋𝑵𝒅𝒚 = 𝟎
Steps in Solving a Linear Equation of Order
Step 1. Write the differential equation 𝑀𝑑𝑥 + 𝑁𝑑𝑦 = 0 in standard form:
𝒅𝒚
+
𝒅𝒙
𝑷(𝒙)𝒚 = 𝑸(𝒙) if linear in 𝑦 or
𝒅𝒙
+
𝒅𝒚
𝑷(𝒚)𝒙 = 𝑸(𝒚) if linear in 𝑥.
Step 2. Find the integrating factor 𝜑:
𝝋 = 𝒆∫ 𝑷𝒅𝒙 if linear in 𝑦 or 𝝋 = 𝒆∫ 𝑷𝒅𝒚 if linear in 𝑥
Step 3. Obtain the general solution:
𝒚𝝋 = ∫ 𝝋𝑸(𝒙)𝒅𝒙 + 𝑪 if linear in 𝑦
𝒙𝝋 = ∫ 𝝋𝑸(𝒚)𝒅𝒚 + 𝑪 if linear in 𝑥