Explain what is the difference between implicit and explicit solutions for differential equation initial value problems.

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Nathaniel Kramer · Accepted Answer

You are attempting to solve for the function y=f when resolving a differential equation (x). An equation of the kind would be a clear answer. If you discover an equation with only two variables, x and y, but you are unsure of how to solve for y, then you have found an implicit answer. After doing your study, you could discover that y=sin(y)+x or a similar equation. However, there isn&#039;t an instantly apparent method to reduce this to y=f(x) form. Once there are no more derivatives, this does uniquely define a solution to the IVP.

user_27qwe · Accepted Answer

Step 1:An initial value problem (IVP) for a differential equation involves finding a solution that satisfies both the differential equation and a set of initial conditions. There are two types of solutions for IVPs: implicit solutions and explicit solutions.Implicit&amp;nbsp;Solution:An implicit solution is represented by an equation in which the dependent variable and its derivatives are related in a non-explicit form. In other words, the dependent variable is not expressed explicitly as a function of the independent variable. Instead, the equation may involve the dependent variable and its derivatives on both sides, without isolating the dependent variable on one side.For example, consider the first-order ordinary differential equation:dydx=f(x,y)An implicit solution to this equation would be an equation of the form:F(x,y)=Cwhere F is a function involving x, y, and their derivatives, and C is a constant determined by the initial conditions. The implicit solution does not explicitly provide the functional relationship between x and y.Explicit&amp;nbsp;Solution:Step 2:In contrast, an explicit solution expresses the dependent variable explicitly as a function of the independent variable. It provides a direct relationship between x and y, without involving derivatives of y on the left-hand side of the equation.Using the same example of a first-order ordinary differential equation:dydx=f(x,y)An explicit solution would be of the form:y=g(x)where g is a function that directly expresses y in terms of x. This type of solution allows for a clear and straightforward understanding of the relationship between the independent and dependent variables.

RizerMix · Accepted Answer

An implicit solution for a differential equation is expressed in terms of an equation that relates the dependent variable y and the independent variable x, without explicitly solving for y as a function of x. In other words, the relationship between x and y is not explicitly given. Instead, the equation typically involves both x and y, and it can be difficult or even impossible to isolate y on one side of the equation. Implicit solutions are commonly represented as equations in the form F(x,y)=0, where F is a function that incorporates both x and y.On the other hand, an explicit solution for a differential equation explicitly expresses y as a function of x. It provides a clear relationship between x and y, allowing us to determine the value of y for any given x. Explicit solutions are typically represented in the form y=f(x), where f is a function that directly relates x to y.To summarize, the main distinction between implicit and explicit solutions lies in the way the relationship between x and y is presented. Implicit solutions involve an equation that relates x and y without explicitly isolating y, while explicit solutions express y as a function of x.

Explain what is the difference between implicit and explicit solutions for differential equation initial value problems.

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