Differential Equation Involving Trigonometric Functions

In summary, the conversation discusses solving two differential equations: \frac{dy}{dx} = cos^2 (x) cos^2 (2y) and y' = \frac{x^2}{y(1+x^3)}. The first equation is solved using integration and the solution is checked by differentiating it. The second equation is solved using a substitution and the solution is also checked by differentiating it. The possibility of a different form of the solution is also mentioned.
  • #1
TranscendArcu
285
0

Homework Statement



Solve the differential equation: [itex]\frac{dy}{dx} = cos^2 (x) cos^2 (2y)[/itex]

The Attempt at a Solution


I rewrote the equation

[itex]\frac{dy}{cos^2 (2y)} = sec^2 (2y) = cos^2 (x) dx[/itex]. Then I integrated,
[itex]\frac{tan(2y)}{2} = \frac{1}{2} (x + sin(x)cos(x)) + c[/itex]. Then I solved for y,
[itex]y = \frac{tan^{-1} (x + sin(x)cos(x) + c)}{2}[/itex]

But this isn't the answer my book gives (or at least it doesn't look very similar). Where did I go wrong?
 
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  • #2
This is another problem I'd like my work checked on (there's no book answer for this):

Solve the differential equation: [itex]y' = \frac{x^2}{y(1+x^3)}[/itex]

Rearranging the equation,

[itex]y dy = \frac{x^2 dx}{1+x^3}[/itex]. Integrating, let [itex]u= 1+x^3[/itex],
[itex]\frac{y^2}{2} = \frac{1}{3} \int \frac{1}{u} du = \frac{1}{3} ln|1+x^3| + c[/itex], which implies,
[itex]y = ± \sqrt{ \frac{2}{3} ln|1+x^3| + c }[/itex]
 
  • #3
TranscendArcu said:

Homework Statement



Solve the differential equation: [itex]\frac{dy}{dx} = cos^2 (x) cos^2 (2y)[/itex]

The Attempt at a Solution


I rewrote the equation

[itex]\frac{dy}{cos^2 (2y)} = sec^2 (2y) = cos^2 (x) dx[/itex]. Then I integrated,
[itex]\frac{tan(2y)}{2} = \frac{1}{2} (x + sin(x)cos(x)) + c[/itex]. Then I solved for y,
[itex]y = \frac{tan^{-1} (x + sin(x)cos(x) + c)}{2}[/itex]

But this isn't the answer my book gives (or at least it doesn't look very similar). Where did I go wrong?

Looks OK to me except that you left off the "dy" that should go with sec^2(2y) up above. How does your answer differ from the one in the book? They might have written sin(x)cos(x) as (1/2)sin(2x).

You can always check that what you have is actually a solution. Start with the equation tan(2y) = x + sin(x)cos(x) and differentiate to find dy/dx. Do the same with the book's answer. If one of the solutions doesn't get back to the differential equation, that solution is wrong.
 

1. What are differential equations involving trigonometric functions?

Differential equations involving trigonometric functions are equations that contain derivatives of trigonometric functions, such as sine, cosine, and tangent. These equations are used to describe relationships between variables that change over time.

2. How are trigonometric functions used in differential equations?

Trigonometric functions are used in differential equations to model periodic behavior and describe the motion of objects in circular or oscillating motion. They can also be used to solve problems involving rates of change, such as in physics and engineering.

3. What are common examples of differential equations involving trigonometric functions?

Some common examples of differential equations involving trigonometric functions include the harmonic oscillator equation, the damped oscillator equation, and the pendulum equation. These equations are used in various fields, including physics, engineering, and economics.

4. How are differential equations involving trigonometric functions solved?

Differential equations involving trigonometric functions can be solved using various methods, such as separation of variables, substitution, and the method of undetermined coefficients. These equations can also be solved numerically using computer software, such as Mathematica or MATLAB.

5. What are the applications of differential equations involving trigonometric functions?

Differential equations involving trigonometric functions have numerous applications in fields such as physics, engineering, economics, and biology. They can be used to model the behavior of systems with periodic motion, such as springs and pendulums, and to analyze rates of change in various processes.

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