First-order Differential Equations - 1 First-order...

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1 First-order Differential Equations In the Math for Engineers 1, students already learnt partial derivative and some differential equations application such as implicit, rate of changes and so on. Here, in the first chapter of Math for Engineers 2, students will learn further on how to solve first-order differential equation (ODE). INTRODUCTION Example of differential equation: ? 2 ? ?? 2 + ?? ?? ?? + ? 3𝑥 = 0 The order of a differential equation is given by the highest derivative involved in the equation. Eg: 3 2 2 2 2 4 3 d f d f df x x y dxdy dx dy There are two types of differential equation which is (1) Homogenous DE (2) Non-homogenous DE Eq: 4 0 dx x dt 2 2 4 cos2 0 d y dy x x dx dx
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2 First-order Differential Equations Solution of differential equations There are a few methods to solve first ODE: 1. By direct integration 2. By separating the variables 3. By substitutions 4. By linear equations Method 1: By direct integration If the equation can be arranged in the form ( ) dy f x dx , then the equation can be solved by simple integration. Example 1 2 3 6 5 dy x x dx Example 2 3 5 4 dy x x dx
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3 First-order Differential Equations Example 3 Find the particular solution of the equation 4 x dy e dx , given that y = 3 when x = 0. Method 2: By separating the variables If the given equation is of the form ( , ) dy f x y dx the variable y on the right-hand side prevents solving by direct integration. Therefore, we need to use other methods. We will now consider a method of solution that can often be applied to first-order equations that are expressible in the form ( ) ( ) dy h y g x dx Such first-order equations are said to be separable. Below are some examples of separable
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  • Spring '13
  • fs
  • Derivative, Partial differential equation

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