Differential equation formation

[lionely]

Homework Statement

If α is an arbitrary constant and a a fixed constant show that
xcos α + ysin α = a

is the complete primitive of the equation

(y - xdy/dx)^2 = a^2( 1 + (dy/dx)^2)

Homework Equations

The Attempt at a Solution

FIrst I found the first derivative by differentiating implicitly,

cosα + (dy/dx)sinα = 0

But now I'm looking for something to sub to try and get the differential equation and eliminate the arbitrary constants, but I keep getting into a loop and cancelling everything basically...

Please help.

 

[Saitama]

Homework Statement

If α is an arbitrary constant and a a fixed constant show that
xcos α + ysin α = a

is the complete primitive of the equation

(y - xdy/dx)^2 = a^2( 1 + (dy/dx)^2)

Homework Equations

The Attempt at a Solution

FIrst I found the first derivative by differentiating implicitly,

cosα + (dy/dx)sinα = 0

But now I'm looking for something to sub to try and get the differential equation and eliminate the arbitrary constants, but I keep getting into a loop and cancelling everything basically...

Please help.

Hi lionely!

You have got two equations. Solve for cos(α) and sin(α) and plug them in the expression $\cos^2\alpha + \sin^2\alpha=1$.

 

[lionely]

But if I use those equations I have all I get is cos(α) in terms of x,y and sin(α) or dy/dx. SO when I put it in the identity I still can't get rid of the arbitrary constants.

 

[pasmith]

Homework Statement

If α is an arbitrary constant and a a fixed constant show that
xcos α + ysin α = a

is the complete primitive of the equation

(y - xdy/dx)^2 = a^2( 1 + (dy/dx)^2)

Homework Equations

The Attempt at a Solution

FIrst I found the first derivative by differentiating implicitly,

cosα + (dy/dx)sinα = 0

But now I'm looking for something to sub to try and get the differential equation and eliminate the arbitrary constants, but I keep getting into a loop and cancelling everything basically...

Please help.

You goal is to confirm that the left hand side of the ODE is equal to the right hand side. If you work out y and dy/dx and just substitute them into the ODE then everything cancelling is exactly what you want!

 

[lionely]

But I want to move from the primitive to the differential.

 

[pasmith]

But I want to move from the primitive to the differential.

Starting from the given solution and trying to manipulate it to obtain the given ODE is not going to work. Your solution is linear; the only way to eliminate the constants of integration is to differentiate twice to obtain y'' = 0. That doesn't help you.

There are two approaches you can take:

• Solve the ODE and show that its general solution takes the given form (this can be done; the easiest way is to differentiate both sides with respect to x; unfortunately the resulting 2nd order equation admits solutions which don't satisfy the original first order ODE, and these solutions must be identified and eliminated).
• Substitute the given form into the ODE and show that everything cancels (this is straightforward).

Actually, I suppose that if you're asked to show that a primitive is a complete primitive then you should use the first approach. Otherwise you've just shown that it is a primitive.

 

[lionely]

If I can't move from the primitive to the differential equation, how the heck did the author of the book get it? This thing is just puzzling me.

 

[Saitama]

But if I use those equations I have all I get is cos(α) in terms of x,y and sin(α) or dy/dx. SO when I put it in the identity I still can't get rid of the arbitrary constants.

Let $\cos\alpha=c$ and $\sin\alpha=e$, then you have the following system of linear equations:

$$cx+ey=a$$
$$c+(dy/dx)e=0$$

Solve the above equations for $c$ and $d$.

 

[lionely]

C = (a-ey)/x

and e= -c/(dy/dx) ? I guess.. Then put these in sin^2 a + cos^2a = 1?

 

[Saitama]

C = (a-ey)/x

and e= -c/(dy/dx) ? I guess.. Then put these in sin^2 a + cos^2a = 1?

I meant that find $c$ and $e$ in terms of x,y and dy/dx and then plug them in the identity.