implicit and explicit solution of differential equation lecture 4

FAMILIES OF SOLUTIONS The study of differential equations is similar to that of integral calculus. In some texts a solution is sometimes referred to as an integral of the equation, and its graph is called an integral curve. When evaluating an antiderivative or indefinite integral in calculus, we use a single constant c of integration. Analogously, when solving a first-order differential equation F(x, y, y) 0, we usually obtain a solution containing a single arbitrary constant or parameter c. A solution containing an arbitrary constant represents a set G(x, y, c) 0 of solutions called a one-parameter family of solutions. When solving an nth-order differential equation F(x, y, y,...,y(n)) 0, we seek an n-parameter family of solutions G(x, y, c1, c2, . . . , cn) 0. This means that a single differential equation can possess an infinite number of solutions corresponding to the unlimited number of choices for the parameter(s). A solution of a differential equation that is free of arbitrary parameters is called a particular solution. For example, the one-parameter family y cx x cos x is an explicit solution of the linear first-order equation xyy x2 sin xon the interval (
,
). (Verify.) Figure 1.1.3, obtained by using graphing software, shows the graphs of some of the solutions in this family. The solution y x cos x, the blue curve in the figure, is a particular solution corresponding to c 0. Similarly, on the interval (
,
),y c1ex c2xex is a two-parameter family of solutions of the linear second-order equation y 2yy 0 in Example 1. (Verify.) Some particular solutions of the equation are the trivial solution y 0 (c1 c2 0), y xex (c1 0, c2 1), y 5ex 2xex (c1 5, c2 2), and so on.

[nadiaofficials]

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[sunilchhareda]

Sir m IIT Kanpur se hu but doubt aapne clear kar diya thanks sir👌👌

[pinkisuthar]

Sir agar implicit or explicit solutions diye ho and 1 differential equation di ho to unko kese proof krenge

Pls explain???

[world_of_coding]

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[FaizRazaRP--EE-]

Thanks sir but it's not in detail but again thanks

[tanzeelanoor]

Salam thank u so much sir bhut a6i smj ai h kafi points clear ho gy h ya differential equations di hovi h an ka graph kasy draw krtay h or intervals k bray m b plzz ak vadio Bana dy Allah apko as ka ajar dy Ameeeen 🤲🤲🤲