Hi can some one help, explain the following please. I seem to have issues with this:
1- Find the corresponding particular solution (in implicit form) that satisfies the initial; condition y = when x = 0.
2- Find the explicit form of this particular solution.
3- What is the value of y given by this particular solution when x =1? Give your answer to four decimal places.
regards Claire
Take the example of a differential equation:
Since the differential equation in question has not been posted, I will work with an example.
y"-2y'+2y=e2x.....(1)
The solution yh to the homogeneous equation
y"-2y'+2y=0 .... (2)
is
yh=C1*ex+C2*x*ex
Note the two integration constants.
To obtain the general solution to equation (1), we must add to yh, solution of (2) a particular solution that satisfies (1). This is called a particular solution, yp.
For this particular problem,
yp=x²ex
which can be obtained by the method of variation of parameters.
Therefore the general solution of (1) is the sum of yh and yp, i.e.
y = C1*ex + C2*x*ex + x²ex ...(general solution)
However, y still contains the constants C1 and C2 to be determined. They can be found by the initial conditions, such as y(0)=0 and y'(0)=1.
Substitute (0,0) into the general solution, we obtain an equation in terms of C1 and C2.
Differentiate the general solution once and substitute x=0, y'(0)=1, we get another equation. From these equations, we can solve for C1 and C2 which is the solution to the problem.
The above general solution is explicit because y, and only y, appears only on the left hand side.
A solution of the form
xy²+x²=2y²
is called an implicit solution because y appears more than once in an equation.
Sure, I can help you with that! It seems like you need assistance with finding the particular solution to a given initial condition, and then finding its explicit form and evaluating it at a specific value. Let's break it down step by step:
1. Finding the corresponding particular solution (implicit form):
To find the particular solution, we first need to have the differential equation provided. Could you please provide me with the differential equation?
2. Finding the explicit form of the particular solution:
Once we have the differential equation and the particular solution in implicit form, we can try to solve for y explicitly. This typically involves using algebraic techniques to isolate y on one side of the equation. Once we obtain an explicit form for y, we will have the particular solution in terms of x.
3. Evaluating the solution at x = 1:
To determine the value of y when x = 1, we substitute x = 1 into the explicit form of the particular solution we found in step 2. This will give us the numerical value of y at x = 1, which can be expressed to four decimal places.
Once you provide me with the differential equation, I can assist you further in finding the particular solution and its explicit form.