What do you mean by general solution and particular solution of a differential equation?

What do you mean by general solution and particular solution of a differential equation?

If the number of arbitrary constants in the solution is equal to the order of the differential equation, the solution is called as the general solution. If the arbitrary constants in the general solution are given particular values, the solution is called a particular solution (of the differential equation).

How do you find the specific solution of a first order differential equation?

Steps

1. Substitute y = uv, and.
2. Factor the parts involving v.
3. Put the v term equal to zero (this gives a differential equation in u and x which can be solved in the next step)
4. Solve using separation of variables to find u.
5. Substitute u back into the equation we got at step 2.
6. Solve that to find v.

What is the basic condition for a function to be a solution to a de?

A solution to a differential equation is a function y=f(x) that satisfies the differential equation when f and its derivatives are substituted into the equation.

How do you find the general solution of an equation?

follow these steps to determine the general solution y(t) using an integrating factor:

1. Calculate the integrating factor I(t). I ( t ) .
2. Multiply the standard form equation by I(t). I ( t ) .
3. Simplify the left-hand side to. ddt[I(t)y]. d d t [ I ( t ) y ] .
4. Integrate both sides of the equation.
5. Solve for y(t). y ( t ) .

What is the general solution of a de?

A solution of a differential equation is an expression for the dependent variable in terms of the independent one(s) which satisfies the relation. The general solution includes all possible solutions and typically includes arbitrary constants (in the case of an ODE) or arbitrary functions (in the case of a PDE.)

What is called particular solution?

[pər¦tik·yə·lər sə′lü·shən] (mathematics) A solution to an ordinary differential equation obtained by assigning numerical values to the parameters in the general solution. Also known as particular integral.

How to find particular solution?

To find the particular solution, you simply take your general solution and plug in the values that you are given for the particular solution. Your general solution is $$y=Ae^x+Be^{2x}+2sin x+6cos x.$$ You have given that the particular solution has the properties $y(0)=0$ and $frac{dy}{dx}(0)=0$.

How to solve differential equations?

Put the differential equation in the correct initial form,(1).

Find the integrating factor,μ(t),using (10).

Multiply everything in the differential equation by μ(t) and verify that the left side becomes the product rule (μ(t)y(t)) ′ and write it as such.

Integrate both sides,make sure you properly deal with the constant of integration.

Solve for the solution y(t).

What are first order differential equations?

A first order differential equation is an equation involving the unknown function y, its derivative y’ and the variable x. We will only talk about explicit differential equations.

What is differential operator?

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.