Solving a Nonlinear Differential Equation with Multiple Methods

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In summary, the conversation discusses solving a non-linear differential equation with the presence of a cosine function. The person is seeking help and has attempted various methods, including linear, exact, Bernoulli, Cauchy, and Legendre, but has not been successful. They are asking for hints or help in solving the equation.
  • #1
abrowaqas
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Homework Statement



Solve the follwing differential equation
2y'-(x/y)+x^3 cosy = 0 solve?

Homework Equations



Linear Differential equation: y'+py=q
exact differential equation: Mdx+Ndy=0

The Attempt at a Solution



both of the methods i have applied ... but didnt work
later i tried it as Bernoulli Differential equation in x... but variables are not coming in

x' + px = q form...


kindly help me or give me hint to solve the above differential equation.
 
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  • #2
This is a non-linear ODE because of cos y. Standard linear ODE solution techniques won't work.
 
  • #3
Give me some hint ... I have also tried Cauchy and legendre methods as well ... But no one works here
 

1. What is the first step in solving this equation?

The first step is to combine like terms and isolate the variable terms on one side of the equation.

2. How do I eliminate the trigonometric function from the equation?

To eliminate the trigonometric function, use the trigonometric identity cos^2(x) + sin^2(x) = 1 to rewrite the equation in terms of cosine only, then use the property cos(x)/cos(y) = 1/tan(y) to eliminate the cosine term.

3. Is there a specific method for solving this type of equation?

Yes, this equation is a nonlinear equation and can be solved using various methods such as substitution, elimination, or graphing. It may also require factoring or the use of the quadratic formula.

4. How many solutions does this equation have?

The number of solutions depends on the values of x and y. Generally, nonlinear equations have multiple solutions, but some values of x and y may result in no real solutions.

5. Can this equation be solved analytically?

Yes, this equation can be solved analytically using algebraic techniques. However, for some values of x and y, it may be more efficient to use numerical methods such as Newton's method or the bisection method.

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