- #1
rapwaydown
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how to solve dy/dx=3√(xy) for a general solution?
PS: √ is square root sign, just in case if it is not clear.
Thank You
PS: √ is square root sign, just in case if it is not clear.
Thank You
csprof2000 said:Nope.
K.J.Healey said:you have to integrate both sides simultaneously.
csprof2000 said:I accidentally did
dy/dx = 3 / sqrt(xy).
I'm surprised you didn't catch this. Oh well. Sorry for the confusion. Your problem is actually easier than the one I did.
dy/dx = 3sqrt(xy)
<=>
dy/sqrt(y) = 3sqrt(x)dx
<=>
2sqrt(y) = 2x^(3/2) + C
<=>
y = [x^(3/2)+C]^2
Hopefully that answers your question. It's separable.
csprof2000 said:A professor, but thankfully not of anything as hard as this. They have me doing the introductory sequence, mostly...
The general approach to solving this type of differential equation is to first separate the variables, then integrate both sides, and finally solve for the dependent variable (in this case, y).
Yes, separation of variables is a commonly used method for solving this type of differential equation. It involves isolating the dependent and independent variables on opposite sides of the equation.
The key steps in solving this differential equation using separation of variables are: 1) Isolate the dependent and independent variables on opposite sides of the equation, 2) Integrate both sides with respect to their respective variables, 3) Rearrange the equation to solve for y, and 4) Substitute any initial conditions to find the specific solution.
Yes, there are other methods such as substitution, integrating factors, and power series solutions. However, for this specific type of differential equation, separation of variables is the most commonly used and efficient method.
This type of differential equation can be used to model growth and decay in biological systems, chemical reactions, and population dynamics. It can also be used in physics to describe the movement of particles in a medium with resistance or the flow of fluids in a porous medium.