{"id":799,"date":"2025-06-20T17:13:30","date_gmt":"2025-06-20T17:13:30","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&p=799"},"modified":"2025-07-22T14:28:34","modified_gmt":"2025-07-22T14:28:34","slug":"basics-of-differential-equations-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/basics-of-differential-equations-learn-it-2\/","title":{"raw":"Basics of Differential Equations: Learn It 2","rendered":"Basics of Differential Equations: Learn It 2"},"content":{"raw":"

Order of Differential Equations<\/h2>\r\n

When working with differential equations, we need a way to categorize and describe them. The most fundamental characteristic is the order<\/strong> of the equation.<\/p>\r\n\r\n

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order of a differential equation<\/h3>\r\nThe order<\/strong> of a differential equation is the highest order of any derivative of the unknown function that appears in the equation.\r\n\r\n<\/section>\r\n

Here are some examples to illustrate:<\/p>\r\n\r\n

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  • [latex]y' = 2x[\/latex] is first-order<\/strong> (highest derivative is [latex]y'[\/latex])<\/li>\r\n \t
  • [latex]y'' - 3y' + 2y = 0[\/latex] is second-order<\/strong> (highest derivative is [latex]y''[\/latex])<\/li>\r\n \t
  • [latex]y''' + xy' = \\sin(x)[\/latex] is third-order<\/strong> (highest derivative is [latex]y'''[\/latex])<\/li>\r\n<\/ul>\r\n

    Understanding the order helps us choose appropriate solution methods and tells us important information about the nature of the solutions we can expect.<\/p>\r\n\r\n

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    What is the order of each of the following differential equations?<\/p>\r\n\r\n

      \r\n \t
    1. [latex]{y}^{\\prime }-4y={x}^{2}-3x+4[\/latex]<\/li>\r\n \t
    2. [latex]{x}^{2}y\\text{'''}-3xy\\text{''}+x{y}^{\\prime }-3y=\\sin{x}[\/latex]<\/li>\r\n \t
    3. [latex]\\frac{4}{x}{y}^{\\left(4\\right)}-\\frac{6}{{x}^{2}}y\\text{''}+\\frac{12}{{x}^{4}}y={x}^{3}-3{x}^{2}+4x - 12[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n[reveal-answer q=\"44558897\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558897\"]\r\n
      \r\n
        \r\n \t
      1. The highest derivative in the equation is [latex]{y}^{\\prime }[\/latex], so the order is [latex]1[\/latex].<\/li>\r\n \t
      2. The highest derivative in the equation is [latex]y\\text{'''}[\/latex], so the order is [latex]3[\/latex].<\/li>\r\n \t
      3. The highest derivative in the equation is [latex]{y}^{\\left(4\\right)}[\/latex], so the order is [latex]4[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>
        Watch the following video to see the worked solution to the example above.\r\n\r\n