The document discusses linear differential equations, specifically first-order equations in the form y′(x) + p(x)y = q(x). It provides examples, including solving for y in specific equations and a practical scenario involving the concentration of salt in a tank. The content is part of a lecture series focused on the topic of differential equations.

1. DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM
DIFFERENTIAL EQUATION (MT-202)
LECTURE #7
LINEAR DIFFERENTIAL EQUATION:
A first order differential equation is linear if it has the form
𝑦′(𝑥) + 𝑃(𝑥)𝑦 = 𝑄(𝑥)
EXAMPLE#1:
𝑑𝑦
𝑑𝑥
+ 𝑦 = 𝑠𝑖𝑛𝑥
EXAMPLE#2: Solve
𝑑𝑦
𝑑𝑥
= 3𝑥2
−
𝑦
𝑥
EXAMPLE#3: Suppose a tank contain 200 gallons of brine (salt mixed with water),
in which 100 pounds of salt is dissolved. A mixture consisting of 1/8 pounds of salt
per gallon is flowing into the tank at a rate of 3 gallon per minute, and the mixture
is continuously stirred. Meanwhile brine is allowed to empty out of the tank at the
same rate of 3 gallons per minute. How much salt is in the tank at any time?