Families of Solutions (Level 1) The study of differential equations is in some respects run similar to those of your calculus II (integral account). As a refresher course, remember that when evaluating an anti-derivative or indefinite integral, we have to remember to close the constant C of integration. An antiderivative that contains the constant of integration represents a whole family of anti-derivatives, one antiderivative for each unique value of the constant C.
For example if we are asked to find the antiderivative or indefinite integral of the quantity 3x ^ 2-1 dx, we would obtain the following antiderivative. x ^ 3 – x + C This antiderivative represents a family of functions and is usually referred to as the general antidervant of the function 3x ^ 2-1. this common antiderivative represents the family of all anti-derivatives of the function 3x ^ 2-1, see that each antiderivative will differ by a constant, it means that the graphs of any two anti-derivatives of the function 3x ^ 2-1 are vertical translations of each other. If we use the antiderivative with chart constant of integration equal to 0 we would obtain the following curve, in the same way If we allow the constant of integration to be equal to 1, negative 1, 2 negative 2, 3 and negative 3 we will obtain the following curves. All these curves represent the different antiderviatives of the function 3x ^ 2-1, if you were to take the derivative of any of these curves you will get the function 3x ^ 2-1, as a result of these curves are all part of a happy family that can be represented by the general antiderivative x ^ 3-x + C.
In this sense, an indefinite integral infinite number of anti-derivatives contains corresponding to the unlimited number of choices that the constant of integration C can achieve. A similar concept exists when solving differential equations, for example when solving first-order differential equations, we usually obtain a solution with a single arbitrary constant which we will call the parameter c. represents a solution with an arbitrary constant a set of solutions known as a one-parameter family of solutions. For example the following function y = ce ^ x represents a one-parameter family of solutions for the differential equation y '- y = 0 on the interval negative infinity to positive infinity, this is a one parameter family of solutions, because it contains one constant, in this case c. In general, when solving a de-order differential equation, we look for an N-parameter family of solutions. As a result, a single differential equation can possess an infinite number of solutions corresponding to the unlimited number of choices for the parameters.
As an additional, for example the following function y = c sub 1 times e ^ x + c sub 2 times e ^ -x, represents a two-parameter family of solutions for the differential equation y '' – y = 0, this is a two-parameter family of solutions, because it contains two constants c sub 1 and c sub 2. Let's talk about the different types of solutions that exist when solving ODE's. A solution of a differential equation that is free of arbitrary parameters is known as a particular solution. For example, if we were to take the one-parameter family of solutions y = ce ^ x and assign values to the parameter c we would obtain the following solution curve. If the parameter c = 0 we will get the next graph, see that this graph represents the trivial solution y = 0 the equator equation y '- y = 0, in the same way we would get the following graphs when c = 1, -1, 2, -2, 3, and -3.
Each of these graphs is particular solutions of the family y = ce ^ x on the interval negative infinity to positive infinity. Keep in mind that a particular solution free of arbitrary parameters. In the same way the two parameter family of solutions also contains infinitely many specific solutions such as the trivial solutions y = 0 when both c sub 1 and c sub 2 are equal to 0. y = e ^ x when c sub 1 is equal to 1 and c sub 2 is equal to 0, the soltuion y = e ^ -x when c sub 1 is equal to 0 and c sub 2 is equal to 1, the solution y = 5th ^ x-2nd ^ -x when c sub 1 is equal to 5 and c sub 2 equals -2, and so on. A family of solutions with N essential parameters are known as a general solution. where N is the order of the differential equation Roughly speaking, solving a nth order differential equation requires N integration and thus leads to N parameters.
For linear ODE's every possible detail solution can be found by appropriate selection of parameters of the general solution; therefore the general solution is also a complete solution. This is not the case for nonlinear ODE's. For nonlinear ODEs (with the exception of some first-order equations) are usually difficult or impossible to solve in terms of elementary functions. If we happen find a family of solutions or general solution, it is not obvious whether it contains family all solutions. Due to the nature of nonlinear ODEs a non-linear differential equation often has to find a solution that cannot be obtained from the general solution. This solution is called a single solution. For example the solution y = 2 / sqrt (x + c) is a general solution of the differential equation 8y '+ y ^ 3 = 0 on the interval negative c excluding positive infinity, see that the solution y = 0 is also a solution of this differential equation, but y = 0 cannot be obtained by assigning a value to the parameter c from the general solution.
This means that the solution y = 0 is a single solution of the nonlinear differential equation. note that the solution y = 0 is also a trivial solution of the differential equation. When dealing with non-linear ODE's on the lookout for some solutions that can not be obtained from the general solution. They may the trivial solution, but this is not always the case. Finally, a solution for a differential equation can be a piece-wise defined function. For example, the family y = cx ^ 2 is a general solution of the differential equation xy'-2y = 0 on the interval negative infinity to positive infinity, using the general solution we can generates two specific solutions, the first is when we 'set the parameter c to 1, and second is when we 'the parameter c equals negative 1. The solution curve of each graph is represented by the following curves. It turns out that we can actually build a solution by using parts of each specific solution, we can essentially use the function y = -x ^ 2 from negative infinity to 0 exclusive, in other words the values of x is less than zero, in combination with the function y = x ^ 2 of 0 inclusive to positive never ending.
In other words the values of x are greater than or equal to 0. This piece shows defined function can be considered as a particular solution for the differential equation. Note that this piece of way defined function is different from both the solution y = x ^ 2 and y = -x ^ 2 when looking at individually, realize that this piece of way defined function can not be obtained of the family y = cx ^ 2, by a single choice of the parameter c. Rather the solution is built up from the family by choosing c = -1 for x <0 and c = 1 for x> = 0. When building piece way defined solutions of differential equations, the piece way defined function must be differentiable and satisfy the equator equation at the given interval of definition. Just like an explicit or implicit solution of a differential equation. Good to summarize, the process of solving differential equations yields solutions that contains arbitrary parameters.
when the solution is an on-the-order differential equation, we look for An N-parameter family of solutions. A family of solutions with N essential parameters is known as a general solution. A solution of a differential equation that is free from arbitrary parameters are known as a particular solution. A solution that cannot be obtained by allocating a value to the parameters c in general solution is referred to as a single solution. and finally, a solution to a differential equation can be a piece-wise defined function by the constraint the domain of two or more specific solutions. Good in our next video will verify practice family of solutions of differential equations..