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btnh
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I am just supposed to find the general solution, in an explicit form if possible.
y'=xcos^(2)y
Thanks!
y'=xcos^(2)y
Thanks!
btnh said:I am just supposed to find the general solution, in an explicit form if possible.
y'=xcos^(2)y
Thanks!
A separable equation is a type of differential equation where the variables can be separated and solved individually. In other words, the equation can be rearranged so that all terms containing the dependent variable are on one side and all terms containing the independent variable are on the other side.
To solve a separable equation, you first rearrange the equation so that all terms containing the dependent variable are on one side and all terms containing the independent variable are on the other side. Then, you integrate both sides with respect to their respective variables. This will result in a general solution, which can then be solved for any initial conditions given.
The notation y' represents the derivative of the function y with respect to the independent variable, in this case, x. It is also known as the slope or rate of change of y.
The cosine squared term, cos^(2)y, is a function of the dependent variable y. This means that the rate of change of y is not constant and depends on the value of y. This makes the equation a non-linear separable equation, which requires a different approach to solving compared to linear separable equations.
As an example, let's solve the equation y'=x^2+3y. First, we rearrange the equation to get y' - 3y = x^2. Then, we integrate both sides with respect to x, resulting in y - (3/2)y^2 = (1/3)x^3 + C. This is the general solution, which can be solved for specific initial conditions if given.