Hướng Dẫn For what value of k equation has no solution? ?
Mẹo Hướng dẫn For what value of k equation has no solution? Mới Nhất
Bùi Đức Thìn đang tìm kiếm từ khóa For what value of k equation has no solution? được Update vào lúc : 2022-12-14 01:05:13 . Với phương châm chia sẻ Mẹo về trong nội dung bài viết một cách Chi Tiết 2022. Nếu sau khi tham khảo Post vẫn ko hiểu thì hoàn toàn có thể lại phản hồi ở cuối bài để Mình lý giải và hướng dẫn lại nha.3x + y -1 = 0
Nội dung chính Show- Explanation[edit]Types of boundary value problems[edit]Boundary value conditions[edit]Differential operators[edit]Electromagnetic potential[edit]For what value of k the equation has unique solution?For what value of k the system has non trivial solution?What is the value of K infinitely many solutions?
(2k -1)x + (k-1)y – 2k -1 = 0
a1/a2 = 3/(2k -1) , b1/b2 = 1/(k-1), c1/c2 = -1/(-2k -1) = 1/( 2k +1)
For no solutions
a1/a2 = b1/b2 ≠ c1/c2
3/(2k-1) = 1/(k -1) ≠ 1/(2k +1)
3/(2k –1) = 1/(k -1)
3k -3 = 2k -1
k =2
Therefore, for k = 2 the given pair of linear equations will have no solution.
The trick to this question is that you want the top equation and the bottom equation to be multiples of eachother of eachother, but having different answers.
That might sound confusing, let me explain!
We want the bottom equation to mirror the top equation.
3x - 7y = 8
6x+ky=-6
We see that the 6x on the bottom equation is twice the amount of the 3x on the top equation. So, we take that 2 and multiply it by the -7y.
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.
Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator.
To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed.
Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's principle.
Explanation[edit]
Boundary value problems are similar to initial value problems. A boundary value problem has conditions specified the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified the same value of the independent variable (and that value is the lower boundary of the domain, thus the term "initial" value). A boundary value is a data value that corresponds to a minimum or maximum input, internal, or output value specified for a system or component.
For example, if the independent variable is time over the domain [0,1], a boundary value problem would specify values for y(t)displaystyle y(t)
both t=0displaystyle t=0
and t=1displaystyle t=1
, whereas an initial value problem would specify a value of y(t)displaystyle y(t) and y′(t)displaystyle y'(t)
time t=0displaystyle t=0.Finding the temperature all points of an iron bar with one end kept absolute zero and the other end the freezing point of water would be a boundary value problem.
If the problem is dependent on both space and time, one could specify the value of the problem a given point for all time or a given time for all space.
Concretely, an example of a boundary value problem (in one spatial dimension) is
y″(x)+y(x)=0displaystyle y''(x)+y(x)=0to be solved for the unknown function y(x)displaystyle y(x)
with the boundary conditionsy(0)=0, y(π/2)=2.displaystyle y(0)=0, y(pi /2)=2.Without the boundary conditions, the general solution to this equation is
y(x)=Asin(x)+Bcos(x).displaystyle y(x)=Asin(x)+Bcos(x).From the boundary condition y(0)=0displaystyle y(0)=0
one obtains
0=A⋅0+B⋅1displaystyle 0=Acdot 0+Bcdot 1which implies that B=0.displaystyle B=0.
From the boundary condition y(π/2)=2displaystyle y(pi /2)=2
one finds2=A⋅1displaystyle 2=Acdot 1and so A=2.displaystyle A=2.
One sees that imposing boundary conditions allowed one to determine a unique solution, which in this case isy(x)=2sin(x).displaystyle y(x)=2sin(x).Types of boundary value problems[edit]
Boundary value conditions[edit]
Finding a function to describe the temperature of this idealised 2D rod is a boundary value problem with Dirichlet boundary conditions. Any solution function will both solve the heat equation, and fulfill the boundary conditions of a temperature of 0 K on the left boundary and a temperature of 273.15 K on the right boundary.
A boundary condition which specifies the value of the function itself is a Dirichlet boundary condition, or first-type boundary condition. For example, if one end of an iron rod is held absolute zero, then the value of the problem would be known that point in space.
A boundary condition which specifies the value of the normal derivative of the function is a Neumann boundary condition, or second-type boundary condition. For example, if there is a heater one end of an iron rod, then energy would be added a constant rate but the actual temperature would not be known.
If the boundary has the form of a curve or surface that gives a value to the normal derivative and the variable itself then it is a Cauchy boundary condition.
Examples[edit]Summary of boundary conditions for the unknown function, ydisplaystyle y
, constants c0displaystyle c_0
and c1displaystyle c_1
specified by the boundary conditions, and known scalar functions fdisplaystyle f
and gdisplaystyle g
specified by the boundary conditions.NameForm on 1st part of boundaryForm on 2nd part of boundaryDirichlety=fdisplaystyle y=fNeumann∂y∂n=fdisplaystyle partial y over partial n=fRobinc0y+c1∂y∂n=fdisplaystyle c_0y+c_1partial y over partial n=fMixedy=fdisplaystyle y=fc0y+c1∂y∂n=fdisplaystyle c_0y+c_1partial y over partial n=fCauchyboth y=fdisplaystyle y=f and c0∂y∂n=gdisplaystyle c_0partial y over partial n=gDifferential operators[edit]
Aside from the boundary condition, boundary value problems are also classified according to the type of differential operator involved. For an elliptic operator, one discusses elliptic boundary value problems. For a hyperbolic operator, one discusses hyperbolic boundary value problems. These categories are further subdivided into linear and various nonlinear types.
Electromagnetic potential[edit]
In electrostatics, a common problem is to find a function which describes the electric potential of a given region. If the region does not contain charge, the potential must be a solution to Laplace's equation (a so-called harmonic function). The boundary conditions in this case are the Interface conditions for electromagnetic fields. If there is no current density in the region, it is also possible to define a magnetic scalar potential using a similar procedure.
For what value of k the equation has unique solution?
Therefore for k=0 , the system of equations have unique solution.For what value of k the system has non trivial solution?
If this determinant is zero, then the system has either no non-trivial solution or an infinite number of solutions.What is the value of K infinitely many solutions?
Hence, the given system of equations will have infinitely many solutions, if k=2. Tải thêm tài liệu liên quan đến nội dung bài viết For what value of k equation has no solution?