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DIFFERENTIAL EUQATIONS
Order Methods of Solving STEP 1 STEP 2
is the order of the highest order derivative in the equation.
0 = 2 0 = 3 0= 2 Variable Separable Homogenous D.E. Linear D.E
STEP 3 STEP 4 Solve



Variable Separable =• 𝑥−𝑦
𝑑𝑦
𝑑𝑥
= 𝑥 + 2𝑦

𝑑𝑦 𝑥+𝑦
Degree • =
𝑑𝑦 𝑑𝑦 1+𝑦 2 𝑑𝑥 𝑥
• = 𝑥𝑦 • = 𝑥 𝑥
𝑑𝑥 𝑑𝑥 1+x2
• 2𝑦𝑒 𝑦 𝑑𝑥 + 𝑦 − 2𝑥𝑒 𝑦 𝑑𝑦 = 0, 𝑤ℎ𝑒𝑛 𝑥 = 0, 𝑦 = 1
is the degree (power) of the highest order derivative 𝑑𝑦 𝑥 𝑑𝑦 𝑥+1
• =𝑦 • = 2−𝑦
𝑑𝑥 𝑑𝑥
0 =3 0 = 3
D =2 D=1 -




Linear D.E
𝑑𝑦 2
Find particular solution when y=1, x=0, = −4𝑥𝑦
𝑑𝑥
𝒅𝒚
0 =2 not defined
.

+ 𝑷𝒚 = 𝑸
The degree of the differential equation
𝑑2 𝑦
+ 𝑠𝑖𝑛
𝑑𝑦 Be

+ 2𝑦 = 0 is 𝒅𝒙 P and Q are constants or functions of x only
𝑑𝑥 2 𝑑𝑥

Homogenous D.E. Integrating Factor (I.F)=𝑒 ‫𝑥𝑑𝑃 ׬‬
Particular & General Solution
• A function F(x, y) is said to be homogeneous function
of degree n if F(λx, λy) = λn F(x, y) for any constant λ. Solution: 𝒚 𝑰. 𝑭 = ‫ 𝒙𝒅 𝑭𝑰 × 𝑸 ׬‬+ 𝑪
• Particular Solution – No Arbitrary Constants
• General Solution –# Arbitrary Constants = Order of D.E • A function is said to be homogenous, if 𝑑𝑦
• 𝐹𝑖𝑛𝑑 𝐼𝐹: 𝑥 𝑑𝑥 + 2𝑦 = 𝑥 2
𝑦 𝑥
𝐹 𝑥, 𝑦 = 𝑥 𝑛 . 𝑔 𝑜𝑟 𝑦 𝑛 . 𝑔
𝑥 𝑦 𝑑𝑦 𝑦
• 𝐹𝑖𝑛𝑑 𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑠𝑜𝑙𝑛: 𝑑𝑥 + 𝑥 = 𝑥2

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