short notes module 7 diff eq

SHORT NOTES MODULE 7 : DIFFERENTIAL EQUATIONS

TERMINOLOGY The highest order of derivative in the differential equation.

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ďƒŚ d3 y ďƒś ďƒŚ d2 y ďƒś dy − 3 ďƒ§ 3 ďƒˇ + x ďƒ§ 2 ďƒˇ = 4y ďƒ§ dx ďƒˇ ďƒ§ dx ďƒˇ dx ďƒ¨ ďƒ¸ ďƒ¨ ďƒ¸ Order - 3

The power of the highest order of derivative in the differential equation.

2

3

ďƒŚ d3 y ďƒś ďƒŚ d2 y ďƒś dy − 3 ďƒ§ 3 ďƒˇ + x ďƒ§ 2 ďƒˇ = 4y ďƒ§ dx ďƒˇ ďƒ§ dx ďƒˇ dx ďƒ¨ ďƒ¸ ďƒ¨ ďƒ¸

Degree - 2

DEGREE

â…† ORDER

SOLVING DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS FIRST ORDER SEPARABLE

FIRST ORDER LINEAR

x-terms and y terms can be separated â…†đ?‘Ś đ?‘“ đ?‘Ľ â…†đ?‘Ś = đ?‘œđ?‘&#x; =đ?‘“ đ?‘Ľ đ?‘” đ?‘Ś â…†đ?‘Ľ đ?‘” đ?‘Ś â…†đ?‘Ľ

standard equation: â…†đ?‘Ś + đ?‘Śđ?‘ƒ đ?‘Ľ = đ?‘„ đ?‘Ľ â…†đ?‘Ľ

LHS : y-term RHS : x-term

identify P(x)

integrate both sides LHS : w.r.t y RHS : w.r.t x

find integrating factor: đ??źđ??š = â…‡ ‍ đ?‘Ľ đ?‘ƒ ׏‏ⅆđ?‘Ľ

put y as subject (if possible)

multiply IF to both sides of standard equation

simplify LHS as

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đ??źđ??š. đ?‘Ś

integrate both sides w.r.t x and simplify

put y as a subject (if possible)