The document discusses higher order differential equations. It defines a higher order differential equation as an equation containing dependent and independent variables and two or more derivatives of the dependent variable with respect to one or more independent variables. It provides hints on solving techniques based on whether the roots are real and equal, real and unequal, or complex conjugates. It then works through an example problem of solving the second order differential equation d2y/dx2 - 3dy/dx + 2y = 0, finding the general solution to be y = 4ex - 2e2x.

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2. Md. Al-Amin
ID: 172015031
Department: CSE
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3. Problem & Solution
(Higher Order Differential Equations)
From-
Ordinary & Partial Differential Equations
(academic purpose book)
Course ode- (MATH-103)
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4. Higher Order Differential Equations
Higher Differential Equation-
A higher order differential equation is an equation containing
dependent and independent variables and two or more derivatives
of the dependent variable with respect to one or more independent
variable.
Solve Hints:
If the Roots are Real and Equal – ( 𝑐1 + 𝑐2^x)𝑒 𝑚𝑥
If the Roots are Real and Unequal - ( 𝑐1 𝑒 𝑚1 𝑥
+ 𝑐2 𝑒 𝑚2 𝑥
)
If the Roots are Complex conjugate - 𝑒 𝑎𝑥[𝐴 cos 𝑏𝑥 + 𝐵 sin 𝑏𝑥] 4

5. 𝑑2 𝑦
𝑑𝑥2 − 3
𝑑𝑦
𝑑𝑥
+ 2𝑦 = 0, x=0, y=2,
𝑑𝑦
𝑑𝑥
= 0
Problem
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6. 𝒅 𝟐 𝒚
𝒅𝒙 𝟐
− 𝟑
𝒅𝒚
𝒅𝒙
+ 𝟐𝒚 = 𝟎 −− −(𝟏)
Let, y = 𝑒 𝑚𝑥
be the trail solution.
∴
𝑑𝑦
𝑑𝑥
= 𝑚𝑒 𝑚𝑥 (diff.w.r.to x)
∴
𝑑2 𝑦
𝑑𝑥2 = 𝑚2 𝑒 𝑚𝑥 (again diff.w.r.to x)
From equation (1)-
𝑚2 𝑒 𝑚𝑥 - 3𝑚𝑒 𝑚𝑥 + 2𝑚𝑒 𝑚𝑥 =0
⇒ 𝑒 𝑚𝑥 (𝑚2 − 3𝑚 + 2) = 0
⇒ 𝑚2 − 3𝑚 + 2 = 0
⇒ 𝑚2
− 2𝑚 − 𝑚 + 2 = 0
⇒ m (m -2) -1 (m -2) = 0
⇒ (m -2) (m-1) = 0
∴ m = 1, 2
Solution
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7. Roots are real and unequal.
So the general solution is-
y = 𝑐1 𝑒 𝑚1 𝑥 + 𝑐2 𝑒 𝑚2 𝑥
∴ y = 𝑐1 𝑒 𝑥
+ 𝑐2 𝑒2𝑥
−− −(2)
⇒ 2 = 𝑐1 𝑒0 + 𝑐2 𝑒0 (∵ 𝑥 = 0)
⇒ 2 = 𝑐1 + 𝑐2 −− − 3
Now from equation (2) –
𝑑𝑦
𝑑𝑥
= 𝑐1 𝑒 𝑥 + 2𝑐2 𝑒2𝑥
⇒ 0 = 𝑐1 𝑒0
+ 2𝑐2 𝑒0
⇒ 0 = 𝑐1 + 2𝑐2 − − − (4)
Now (3) – (4)
2 = 𝑐1 + 𝑐2
0 = 𝑐1 + 2𝑐2
2 = − 𝑐2
⇒ 𝑐2 = −2
From equation (3)
2 = 𝑐1 + −2
⇒ − 𝑐1= − 2 − 2
⇒ 𝑐1= 4
Hence the general
solution is –
y = 4𝑒 𝑥 - 2𝑒2𝑥 (Ans:) 7

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