MATH V23 LECTURE NOTES (Bowen) Section 2.2 Separable Equations Unless otherwise stated, it will be assumed in this section that is the independent variable, and x is the dependent variable, for each differential equation discussed. y At long last, we are about to learn (in the next several sections) how to find closed-form solutions to several classes of first-order ODEs. The first method, introduced in this section, relies on the basic integration techniques you learned in MATH V21B. Solution by integration. You may recall from the last section that, for equations of the form if the function is a function of only, the equation is called an autonomous equation. If, f y however, the expression on the right side of the equation is a function of only (), the equation x can be solved by multiplying both sides by which converts the equation to differential form: dx, Provided that has an antiderivative, a family of closed-form solutions may be obtained by simply integrating both sides and to the specific antiderivative. We may extend appending “” this technique to a wider class of functions , however, by noting that integration may also be used if can be rewritten as the product of a function of only and a function of only; that is, f x y if This type of equation is said to be or to have Sometimes it is separable, separable variables. not initially obvious that an equation is separable. For example, consider the ODE in which it appears that the and variables are hopelessly entangled with each other. x y However, by consulting a , we find (using one of the product-to-sum formulas) that the expression on the right side of the equals sign may be rewritten as which is a separable equation, with and . To solve this new equation, we start by using algebra to isolate all factors containing or on the left side, and all factors containing or on the y dy x dx right side. Multiplying both sides by yields the differential forms
MATH V23 LECTURE NOTES (Bowen) Section 2.2 Separable Equations Unless otherwise stated, it will be assumed in this section that x is the independent variable, and y is the dependent variable, for each differential equation discussed. At long last, we are about to learn (in the next several sections) how to find closed-form solutions to several classes of first-order ODEs. The first method, introduced in this section, relies on the basic integration techniques you learned in MATH V21B. Solution by integration. You may recall from the last section that, for equations of the form if the function f is a function of y only, the equation is called an autonomous equation. If, however, the expression on the right side of the equation is a function of x only (), the equation can be solved by multiplying both sides by dx, which converts the equation to differential form: Provided that has an antiderivative, a family of closed-form solutions may be obtained by simply integrating both sides and appending “” to the specific antiderivative. We may extend this technique to a wider class of functions , however, by noting that integration may also be used if f can be rewritten as the product of a function of x only and a function of y only; that is, if This type of equation is said to be separable, or to have separable variables. Sometimes it is not initially obvious that an equation is separable. For example, consider the ODE in which it appears that the x and y variables are hopelessly entangled with each other. However, by consulting a , we find (using one of the product-to-sum formulas) that the expression on the right side of the equals sign may be rewritten as which is a separable equation, with and . To solve this new equation, we start by using algebra to isolate all factors containing y or dy on the left side, and all factors containing x or dx on the right side. Multiplying both sides by yields the differential forms
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