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**Unformatted text preview: **Chapter 2. First Order Diﬀerential Equations §2.3 Linear Equations 1. Def: A ﬁrst order diﬀerential equation of the form a1 (x) dy + a0 (x)y = g (x) dx is said to be a linear equation. When g (x) = 0, the equations is called homogeneous, otherwise it is nonhomogeneous. To solve a linear DE, we rewrite it into its standard form dy + P (x)y = f (x). dx (1) 2. Integrating factor method: The term e P (x)dx is called the integrating factor of the equation. Multiplying equation (1) by it yeilds e
P (x)dx dy dx +e P (x)dx P (x)y = e P (x)dx f (x) which is equivalent to d e dx Integrating gives e and then y = Ce−
P (x)dx P (x)dx P (x)dx y =e P (x)dx f (x). y=C+ e P (x)dx f (x)dx + e− P (x)dx e P (x)dx f (x)dx 3. The Method of Variation of Parameters. The solution of the homogeneous solution is y = Ce−
P (x)dx Now we assume that the solution of equation (1) is y = u(x)e−
P (x)dx = u(x)y1 (x). 1 Pluging it into (1), we can ﬁnd the function u(x) as u(x) = C + EX: Solve the following DEs. (1) (2)
dy dx dy dx e P (x)dx f (x)dx. − 4y = 0 − 4y = 5 dy (3) x dx − 4y = x6 ex (4) dy dx + y = x, y (0) = 4 y (1) = 2. (5) xy + y = ex , 2 ...

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