Dif.eq. with trigonometric functions involving y

[mausmust]

I tried to solve it but confused. Pls. help me to solve this equation:

dy/dx + (e^x)*Sec(y) = Tan(x);

(hint: integrating factor is e^-ax, and a is unknown, a ε ℝ, find it, solve the equation)

Thnx.

 

[tiny-tim]

welcome to pf!

hi mausmust! welcome to pf!

(try using the X² button just above the Reply box )

show us your full calculations, and then we'll see what went wrong, and we'll know how to help!

 

[mausmust]

Hii, thank you :)

I did:

dy + (e^xSec(y)-Tan(x))dx = 0

0 ≠ e^xSec(y)Tan(y) (∂N/∂x ≠ ∂M/∂y) So, we need an integrating factor. If we use e^-ax;

-ae^-ax = e^(1-ax)Sec(y)Tan(y) appears.

Shouldn't be "a" is a real value? Also;

I tried to modify equation with ArcSecant function to become a linear equation with this form;

dy/dx + P(x)y = Q(x)

but it went more complicated. How can we solve it?

 

[jackmell]

I don't think you're doin' that right. Also, don't use capitals letters for the trig functions. If you have:

$$Mdx+Ndy=0$$

and:

$$\frac{1}{N}\left(\frac{\partial M}{\partial y}-\frac{\partial N}{\partial x}\right)=f(x)$$

then the integrating factor is:

$$u=\exp\left(\int f(x)dx\right)$$

but that's not just $e^{-ax}$ for real a.

 

[blue_raver22]

Hello,

Thank you for reaching out for help with this differential equation involving trigonometric functions. I can provide some guidance on how to solve it.

First, let's rewrite the equation in a more standard form:

dy/dx + e^x * sec(y) - tan(x) = 0

To solve this type of equation, we need to find an integrating factor, which is a function that we multiply both sides of the equation by to make it easier to integrate. In this case, the integrating factor is e^-ax, where a is an unknown constant.

To find the value of a, we can use the fact that the left side of the equation should be equal to the derivative of the product of the integrating factor and y. This is known as the product rule in calculus.

So, we have:

(dy/dx + e^x * sec(y) - tan(x)) * e^-ax = d/dx (e^-ax * y)

Expanding the left side of the equation and simplifying, we get:

dy/dx * e^-ax + e^x * sec(y) * e^-ax - tan(x) * e^-ax = d/dx (e^-ax * y)

Next, we can use the chain rule to simplify the right side of the equation:

dy/dx * e^-ax + e^x * sec(y) * e^-ax - tan(x) * e^-ax = e^-ax * (dy/dx - ay)

Now, we can equate the coefficients of dy/dx on both sides:

1 = -ae^-ax

Solving for a, we get a = -1.

Now that we have the integrating factor, we can multiply both sides of the original equation by e^-x:

e^-x * dy/dx + sec(y) - e^-x * tan(x) = 0

Using the chain rule again, we can rewrite the left side as:

d/dx (e^-x * y) + sec(y) - e^-x * tan(x) = 0

Integrating both sides with respect to x, we get:

e^-x * y + ln(sec(y)) + e^-x * (-ln(cos(x))) = C

Where C is the constant of integration.

From here, you can solve for y in terms of x and the constant C