r/learnmath • u/Fun-Note6424 New User • 1d ago

Is my ODE solution reliable enough to submit for the exam?

cos y dx - sin y dy = 0
cos y dx = sin y dy
cos y = sin y dy/dx
exp x) cos y = exp x) sin y dy/dx
exp x) cos y - exp x) sin y dy/dx = 0
(exp(x) cos y)' = 0
int (exp(x) cos y)' dx = 0
exp x) cos y + c = 0
exp x) cos y = c

The reason I am being cautious is that:

  1. I was taught that this method of solving applies only to linear cases; and

  2. I was told that one must be careful when dealing with dy dx and that I am not yet at a level where I can fully grasp those concepts.

1 Upvotes

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3

u/espressoristretto New User 23h ago

Why not just separate the variables? 

int dx = int tan y dy

x+c=ln sec y

Aexp(x)cos(y)=1

2

u/Fun-Note6424 New User 22h ago edited 22h ago

I don't like tan differenting. sin cos twice is hastle.

2

u/Bounded_sequencE New User 21h ago

  1. This method is generally valid -- but only for linear ODE's is it a general solution strategy.

  2. Regarding "dy, dx": If you want to keep it rigorous, don't split them, like in this assignment. Instead, rewrite the ODE as "cos(y) - sin(y)*y' = 0". Multiply by "exp(x)", and use product- and chain-rule of derivatives backwards

    0 = exp(x)cos(y) - exp(x)sin(y)y' = d/dx exp(x)cos(y)

Integrate both sides by "x", and use FTC, just as you did.

1

u/Southlander24 A friendly Redditor!👋 1d ago edited 1d ago

Your method is fine because it just relies on the reverse product rule. You don't need to justify 'dividing' both sides by dx.

The only thing I would change is to rewrite the third-last line as: "int (exp(x) cos y)' dx = int 0 dx". Also, your brackets are misaligned.

3

u/Fun-Note6424 New User 1d ago

Thany kou! Have a great day.

1

u/Southlander24 A friendly Redditor!👋 1d ago

You too!

1

u/MathsFromNothing New User 23h ago

I agree with you to the third line, then I'm not sure what you are doing. The image shows my working and check of the solution using the Maths from Nothing computer algebra system.

The typical approach with this type of ODE is to get all of the y terms on one side of the equation with the dydx and the x terms on the other. The derivative and the integral combine to make it an integral with respect to y and the other side with respect to x. Perform the integration and solve for y.

1

u/MathsFromNothing New User 23h ago

I think this is your method. Your matching coefficients with the product rule. The working using the same CAS.