Question

Find a differential equation whose solution is a family of straight lines that are tangents to the parabola \[\ y^2=2x \]

Any help would be appreciated.

Answer 1

I think you just need to use implicit differentiation here, so:\[y^2=2x\]therefore:\[2y\frac{dy}{dx}=2\]leaving the differential equation:\[y\frac{dy}{dx}=1\]

Answer 2

Hmm, yea but it needs to be a family of straight lines where all the solutions are tangents to the parabola, which I suppose means to take into account only those lines which actually touch the parabola

so it would be \[\ y= kx + C(y')\]
Where
k = y' (for y^2=2x)

I'm having trouble with determining C(y') if it is actually only dependent on y'.

Answer 3

I have a solution and I will post it in a sec

Answer 4

ah - yes - you are right - I didn't read the question properly

Answer 5

\[\ 2x(y')^2 -2yy' +1 =0 \]

Answer 6

Given that the derivative of that formula satisfies yy' = 1, that mean the slope of the tangent is y' = 1/y. To avoid confusion between the original curve and the variables of the straight line, Write this as
\[ y_0' = 1/y_0 \]
Now the equation of the straight line is
\[ y - y_0 = \frac{1}{y_0} (x - x_0) \] and you have \( x_0 = y_0^2/2 \). Hence
\[ y = f(x) = \frac{1}{y_0} (x - \frac{y_0^2}{2} ) \]
i.e.,
\[ f(x) = \frac{x}{y_0} - \frac{y_0}{2} \] Now the only question is: what differential equation would have this as a solution, where \( y_0 \) is a constant that arises naturally in the course of solving the equation.

Answer 7

To find that, find a function g(x,y) = y_0. i.e., solve the quadratic equation in y_0.

Then, finally, find a differential equation that g(x,y) satisfies.

Answer 8

Ok I got it. I didn't solve any quadratic equations.

There is a small error here. It should be \[\ f(x) = y = \frac{x}{y_0} + \frac{y_0}{2} \](plus, not minus)
Since I know that \[\ y_0 = 1/y' \]I plug this in in the above equation and I get the correct result.

Answer 9

ah, right, yes.

Answer 10

Thank you very much :D

Answer 11

2x(y′)2−2yy′+1=0
guys i tried to find how it became like this but still cant find the right path arriving to this equation