Hello, I need help making the following formula equal, y= ... instead of xy=... I know that I could send the x under cx + cy but then what do I do with the cy?
Heres the whole question show that the given equation is a solution of the given differential equation xy' + y^2 = 0 , xy = cx + cy Please help me solve this.
How about just solving for y? Move all terms with y onto one side of the equation and all other terms to the other. Factor out y, divide both sides by the resulting coefficient of y.
Are you sure that these equations are the given equations? Because they are not dimensionally consistent. The right hand equation requires that c, x, and y all have the same units (call them 'fred'). This means that y'=dy/dx is dimensionless. This in turn means that the left hand equation has units of 'fred' for the first term and 'fred'^2 for the second term. Are you sure the diffy-Q isn't (x^2)y' + y^2 = 0
I have no choice but to assume that 'c' is a constant. But what do you mean "as if *they* were constants," there's only one 'c'. If you mean to say that the two occurances are not the same constant, then you need to call them something else. But none of that addresses the dimensional inconsistency.
I have to agree with WBahn, or the answer is that xy=xc + cy it is not a solution of xy' +y^2=0 BTW, y=cx/(x-c)
xy = cx + cy xy - cy = cx + cy - cy xy -cy = cx +0 xy - cy = cx y(x-c) = cx y(x-c)/(x-c) = cx / (x-c) y * 1 = cx/(x-c) y=cx/(x-c) differentiate to get y' = [c(x-c) - cx]/(x-c)^2 y'= -c^2/(x-c)^2 also calculate y^2 y^2=[cx/(x-c)]^2 then try to verify xy' + y^2 = 0 x*[-c^2/(x-c)^2]+[cx/(x-c)]^2 = 0 which is not true except for special case where x=1 because x <> x^2
And hopefully the OP (and others?) will take note that I didn't even have to try to solve the problem to spot (1) that the given equation could not be a solution of the given differential equation, and that (2) what the form of the diffy-Q would need to be in order for it to be a solution. That's part of the power of dimensional analysis -- even when there are no dimensions, per se.
Actually this problem is simple algebra, which is bread and butter to all electronics one way or another. I didn't even consider the dimensional aspects, treating it as pure math. It wasn't until college chemistry I ran into the importance of units, and making sure all the i's were dotted and t's were crossed. Funny thing is it is every bit as true for electronics.
Actually, the problem isn't pure algebra, since it involves a differential equation. But if you treat it as a "pure math problem", then you have little choice but to crank through the math and solve for y, take the derivative, square y, and plug all of that into the differential equation. Then, when you discover that it doesn't work, you are left to wonder whether it really doesn't work, or whether you just made a mistake along the way. So now you spend more time redoing it and/or checking your work to make sure that you didn't make any mistakes. For a simple problem like this, all of that doesn't take too long. But this could easily have been something that took a couple pages to work out (without looking much more complicated at face value than this one). But in either case, if you look for dimensional consistency, even in a 'pure math problem', you can spot things like this without doing any math at all and KNOW that it won't work out. Perhaps I'm just silly, but I prefer to spot problems early whenever possible.
I totally agree with WBahn. I teach my students to use dimensional analysis whenever possible, especially to do a quick check of answers for reasonableness. By the way, I teach dimensional analysis to my honors Algebra I students in September so they have yet one more tool in their toolbox. As students mature in their math knowledge, the applications are endless.