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DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM
DIFFERENTIAL EQUATION (MT-202)
LECTURE #5
TYPE 3 EQUATIONS REDUCIBLE TO HOMOGENOUS:
If equation
𝑑𝑦
𝑑𝑥
= 𝑓(𝑥, 𝑦) is of the form
𝑑𝑦
𝑑𝑥
=
𝐴𝑥 + 𝐵𝑦 + 𝐶
𝐴′ 𝑥+𝐵′ 𝑦 + 𝐶′
(1)
It can be reduced to the homogenous form by substituting𝑥 = 𝑋 + ℎ, 𝑦 = 𝑌 + 𝑘.
Therefore
𝑑𝑦
𝑑𝑥
=
𝑑𝑌
𝑑𝑋
(2)
And the equation (1) reduces to
𝑑𝑌
𝑑𝑋
=
𝐴𝑋 + 𝐵𝑌 + 𝐴ℎ + 𝐵𝑘 + 𝐶
𝐴𝑋′ + 𝐵𝑌′ + 𝐴′ℎ + 𝐵′ 𝑘 + 𝐶′
Now, choose h and k so that 𝐴ℎ + 𝐵𝑘 + 𝐶 = 0 and 𝐴′
ℎ+𝐵′
𝑘 + 𝐶′
= 0 which
gives
ℎ =
𝐵𝐶′
− 𝐶𝐵′
𝐴𝐵′ − 𝐵𝐴′
, 𝑘 =
𝐶𝐴′
− 𝐴𝐶′
𝐴𝐵′ − 𝐵𝐴′
By substituting these values in (2), we get
𝑑𝑌
𝑑𝑋
=
𝐴𝑋 + 𝐵𝑌
𝐴′ 𝑋+𝐵′ 𝑌
Which is a homogenous differential equation and can be solved by the method
discussed in Type 2.
CASE OF FAILURE:
In this case, the differential equation reduces to variable separable type.
If
𝐴
𝐴′
=
𝐵
𝐵′
, then the value of h and k will not exist. In this case, let
𝐴
𝐴′
=
𝐵
𝐵′
=
1
𝑚
=> 𝐴′
= 𝑚𝐴 And 𝐵′
= 𝑚𝐵
Then the given equation becomes
DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM
DIFFERENTIAL EQUATION (MT-202)
𝑑𝑦
𝑑𝑥
=
𝐴𝑥 + 𝐵𝑦 + 𝐶
𝑚(𝐴𝑥 + 𝐵𝑦) + 𝐶′
To solve it first we put 𝐴𝑥 + 𝐵𝑦 = 𝑧 and then apply the method of variable
separable to solve the transformed differential equation.
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differential equation Lecture#5

  • 1. DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM DIFFERENTIAL EQUATION (MT-202) LECTURE #5 TYPE 3 EQUATIONS REDUCIBLE TO HOMOGENOUS: If equation 𝑑𝑦 𝑑𝑥 = 𝑓(𝑥, 𝑦) is of the form 𝑑𝑦 𝑑𝑥 = 𝐴𝑥 + 𝐵𝑦 + 𝐶 𝐴′ 𝑥+𝐵′ 𝑦 + 𝐶′ (1) It can be reduced to the homogenous form by substituting𝑥 = 𝑋 + ℎ, 𝑦 = 𝑌 + 𝑘. Therefore 𝑑𝑦 𝑑𝑥 = 𝑑𝑌 𝑑𝑋 (2) And the equation (1) reduces to 𝑑𝑌 𝑑𝑋 = 𝐴𝑋 + 𝐵𝑌 + 𝐴ℎ + 𝐵𝑘 + 𝐶 𝐴𝑋′ + 𝐵𝑌′ + 𝐴′ℎ + 𝐵′ 𝑘 + 𝐶′ Now, choose h and k so that 𝐴ℎ + 𝐵𝑘 + 𝐶 = 0 and 𝐴′ ℎ+𝐵′ 𝑘 + 𝐶′ = 0 which gives ℎ = 𝐵𝐶′ − 𝐶𝐵′ 𝐴𝐵′ − 𝐵𝐴′ , 𝑘 = 𝐶𝐴′ − 𝐴𝐶′ 𝐴𝐵′ − 𝐵𝐴′ By substituting these values in (2), we get 𝑑𝑌 𝑑𝑋 = 𝐴𝑋 + 𝐵𝑌 𝐴′ 𝑋+𝐵′ 𝑌 Which is a homogenous differential equation and can be solved by the method discussed in Type 2. CASE OF FAILURE: In this case, the differential equation reduces to variable separable type. If 𝐴 𝐴′ = 𝐵 𝐵′ , then the value of h and k will not exist. In this case, let 𝐴 𝐴′ = 𝐵 𝐵′ = 1 𝑚 => 𝐴′ = 𝑚𝐴 And 𝐵′ = 𝑚𝐵 Then the given equation becomes
  • 2. DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM DIFFERENTIAL EQUATION (MT-202) 𝑑𝑦 𝑑𝑥 = 𝐴𝑥 + 𝐵𝑦 + 𝐶 𝑚(𝐴𝑥 + 𝐵𝑦) + 𝐶′ To solve it first we put 𝐴𝑥 + 𝐵𝑦 = 𝑧 and then apply the method of variable separable to solve the transformed differential equation.
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