{"id":800,"date":"2025-06-20T17:13:32","date_gmt":"2025-06-20T17:13:32","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&p=800"},"modified":"2025-07-22T14:33:05","modified_gmt":"2025-07-22T14:33:05","slug":"basics-of-differential-equations-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/basics-of-differential-equations-learn-it-3\/","title":{"raw":"Basics of Differential Equations: Learn It 3","rendered":"Basics of Differential Equations: Learn It 3"},"content":{"raw":"

General Solutions vs. Particular Solutions<\/h2>\r\n

We already noted that the differential equation [latex]y' = 2x[\/latex] has at least two solutions: [latex]y = x^2[\/latex] and [latex]y = x^2 + 4[\/latex]. But there's something important happening here.<\/p>\r\n

The only difference between these solutions is the constant term. What if we tried a different constant? Would [latex]y = x^2 + 7[\/latex] work? How about [latex]y = x^2 - 3[\/latex]?<\/p>\r\n

The answer is yes! Any function of the form [latex]y = x^2 + C[\/latex], where [latex]C[\/latex] represents any constant, is a solution. Here's why: the derivative of [latex]x^2 + C[\/latex] is always [latex]2x[\/latex], regardless of the value of [latex]C[\/latex] (since the derivative of a constant is zero).<\/p>\r\n

It turns out that every solution to this differential equation must have the form [latex]y = x^2 + C[\/latex]. This leads us to two important concepts.<\/p>\r\n\r\n

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general solution vs. particular solution<\/h3>\r\n
    \r\n \t
  • General Solution<\/strong>: A solution that contains all possible solutions to a differential equation. It includes an arbitrary constant (or constants) and represents a family of curves.<\/li>\r\n \t
  • Particular Solution<\/strong>: A specific solution obtained by choosing a particular value for the constant(s) in the general solution.<\/li>\r\n<\/ul>\r\n<\/section>
    \r\n

    For [latex]y' = 2x[\/latex]:<\/p>\r\n\r\n

      \r\n \t
    • General solution: [latex]y = x^2 + C[\/latex] (family of all solutions)<\/li>\r\n \t
    • Particular solution: [latex]y = x^2 - 3[\/latex] (when [latex]C = -3[\/latex])<\/li>\r\n<\/ul>\r\n<\/section>Figure 1 shows this family of solutions graphically. Each curve represents a different value of [latex]C[\/latex], but they all satisfy the same differential equation.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]\"A Figure 1. Family of solutions to the differential equation [latex]{y}^{\\prime }=2x[\/latex].[\/caption]
      Think of it visually: The general solution gives you a whole family of curves. A particular solution picks out just one curve from that family.\r\n\r\n<\/section>Often, we can find a unique particular solution when we're given additional information about the problem\u2014but we'll explore that idea next.\r\n\r\n
      \r\n
      \r\n

      Find the particular solution to the differential equation [latex]{y}^{\\prime }=2x[\/latex] passing through the point [latex]\\left(2,7\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558894\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558894\"]\r\n

      \r\n

      Any function of the form [latex]y={x}^{2}+C[\/latex] is a solution to this differential equation. To determine the value of [latex]C[\/latex], we substitute the values [latex]x=2[\/latex] and [latex]y=7[\/latex] into this equation and solve for [latex]C\\text{:}[\/latex]<\/p>\r\n\r\n

      [latex]\\begin{array}{}\\\\ \\\\ y={x}^{2}+C\\hfill \\\\ 7={2}^{2}+C=4+C\\hfill \\\\ C=3.\\hfill \\end{array}[\/latex]<\/div>\r\n \r\n

      Therefore the particular solution passing through the point [latex]\\left(2,7\\right)[\/latex] is [latex]y={x}^{2}+3[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>

      Watch the following video to see the worked solution to example above.