Solving Differential Equations Involving Trigonometric Functions

When dealing with differential equations involving trigonometric functions, such as (frac{dy}{dx} sin(xy)cos(x)y), we need to employ several techniques to find the solution. This article will guide you through solving such an equation by using a substitution method, specifically the Weierstrass substitution.

Problem Statement: Solving (frac{dy}{dx} sin(xy)cos(x)y)

To solve the differential equation (frac{dy}{dx} sin(xy)cos(x)y), let's start by making a substitution to simplify the equation.

Step 1: Substitution

Let (u xy). Then, we have

[frac{du}{dx} y xfrac{dy}{dx}]

Rearranging and solving for (frac{dy}{dx}), we get:

[frac{dy}{dx} frac{frac{du}{dx} - y}{x}]

Substituting this into the original differential equation:

[frac{frac{du}{dx} - y}{x} sin(u)cos(x)cdot y]

Multiplying both sides by (x), we obtain:

[frac{du}{dx} - y xysin(u)cos(x)]

Since (u xy), we can rewrite the equation as:

[frac{du}{dx} - u sin(u)cos(x)cdot y]

Now, let's simplify further using the Weierstrass substitution:

Step 2: Weierstrass Substitution

Let (v tanleft(frac{u}{2}right)). Then, we have

[sin(u) frac{2v}{v^2 - 1}]

[cos(u) frac{1 - v^2}{v^2 - 1}]

Therefore,

[frac{du}{dx} frac{d}{dx}left(2tan^{-1}(v)right) frac{2}{1 v^2} cdot frac{dv}{dx}]

Substituting these into the equation:

[frac{2}{1 v^2} cdot frac{dv}{dx} - u frac{2v(1 - v^2)}{(v^2 - 1)^2}cos(x)cdot y]

Since (y frac{v^2 - 1}{2v}), the equation becomes:

[frac{2}{1 v^2} cdot frac{dv}{dx} - u frac{2v(1 - v^2)}{(v^2 - 1)^2} cdot cos(x) cdot frac{v^2 - 1}{2v}]

Simplifying, we get:

[frac{2}{1 v^2} cdot frac{dv}{dx} - u frac{1 - v^2}{v^2 - 1} cdot cos(x)]

Finally, integrating both sides, we obtain:

[int frac{2}{1 v^2} cdot dv x C]

Thus,

[2tan^{-1}(v) x C]

Substituting back (v tanleft(frac{u}{2}right)) and (u xy):

[2tan^{-1}left(tanleft(frac{xy}{2}right)right) x C]

This simplifies to:

[lnleft(frac{1 cos(xy)}{1 - cos(xy)}right) x C]

So the final solution is:

[x lnleft(frac{1 cos(xy)}{1 - cos(xy)}right) C]

Conclusion

The solution to the differential equation (frac{dy}{dx} sin(xy)cos(x)y) is given by: