How do you find the general solution?
General Solutions of a Trig EquationSolve sin(x) = y for x.Case 1: -1≤y≤ 1, that is, the value of y is between -1 and 1, so there is a solution.Case 2: -1 > y or y > 1 , that is, the value of y is too large or too small for a solution to be possible.Solve cos(x) = y for x.Case 1: -1≤y≤ 1.Solve tan(x) = y for x.
What is general and particular solution of differential equation?
When the arbitrary constant of the general solution takes some unique value, then the solution becomes the particular solution of the equation. By using the boundary conditions (also known as the initial conditions) the particular solution of a differential equation is obtained.
How do you find the general and singular solution of a differential equation?
Another way to find a singular solution as the envelope of the family of integral curves is based on using C-discriminant. Let Φ(x,y,C) be the general solution of a differential equation F(x,y,y′)=0. Graphically the equation Φ(x,y,C)=0 corresponds to the family of integral curves in the xy-plane.
What is a general solution?
1 : a solution of an ordinary differential equation of order n that involves exactly n essential arbitrary constants. — called also complete solution, general integral. 2 : a solution of a partial differential equation that involves arbitrary functions. — called also general integral.
What is the general solution of tan?
Hence, the general solution of tan θ = tan ∝ is θ = nπ + ∝, where n ∈ Z (i.e., n = 0, ± 1, ± 2, ± 3,…….) Note: The equation cot θ = cot ∝ is equivalent to tan θ = tan ∝ (since, cot θ = 1/tan θ and cot ∝ = 1/tan ∝). Thus, cot θ = cot ∝ and tan θ = tan ∝ have the same general solution.
What is linear equation in differential equation?
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form. where , , and are arbitrary differentiable functions that do not need to be linear, and.
How do you find the general solution of a first order differential equation?
So this: dy dx − y x = 1. Becomes this:u dv dx + v du dx − uv x = 1. Step 2: Factor the parts involving v.Factor v:u dv dx + v( du dx − u x ) = 1. Step 3: Put the v term equal to zero.v term equal to zero: du dx − u x = 0. So: du dx = u x. Step 4: Solve using separation of variables to find u. Separate variables: du u = dx x.
What is differential equation in math?
In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
What is clairaut’s equation of differential equation?
Clairaut’s equation, in mathematics, a differential equation of the form y = x (dy/dx) + f(dy/dx) where f(dy/dx) is a function of dy/dx only. The equation is named for the 18th-century French mathematician and physicist Alexis-Claude Clairaut, who devised it.
What do you mean by exact differential equation?
A first-order differential equation (of one variable) is called exact, or an exact differential, if it is the result of a simple differentiation. The equation P(x, y)y′ + Q(x, y) = 0, or in the equivalent alternate notation P(x, y)dy + Q(x, y)dx = 0, is exact if Px(x, y) = Qy(x, y).
What is an initial value problem in differential equations?
An initial value problem is a differential equation with where is an open set of , together with a point in the domain of , called the initial condition. A solution to an initial value problem is a function that is a solution to the differential equation and satisfies .
What is the general solution of sin theta?
sin θ = sin ∝ θ = nπ + (-1)n ∝, where n ∈ Z. Hence, the general solution of sin θ = sin ∝ is θ = nπ + (-1)n ∝, where n ∈ Z. Note: The equation csc θ = csc ∝ is equivalent to sin θ = sin ∝ (since, csc θ = 1sinθ and csc ∝ = 1sin∝). Thus, csc θ = csc ∝ and sin θ = sin ∝ have the same general solution.