dy/dx-(tan y)/((1+x)*sec y)=(1+x)*e^xThis is a linear differential equation How exactly would you solve this please answer asap

To solve the given linear differential equation, we start with the expression:

dy/dx - (tan y)/((1+x)*sec y) = (1+x)*e^x.

This equation can be rearranged into a standard linear form. A linear first-order differential equation typically looks like:

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

where P(x) and Q(x) are functions of x. In our case, we need to manipulate the equation to fit this format.

Rearranging the Equation

First, let's isolate dy/dx:

dy/dx = (tan y)/((1+x)*sec y) + (1+x)*e^x.

Next, we can simplify the term involving y. Recall that:

tan y = sin y / cos y and sec y = 1 / cos y.

Thus, we can rewrite the equation as:

dy/dx = (sin y / ((1+x)cos y)) + (1+x)e^x.

Identifying P(x) and Q(x)

To express this in the linear form, we need to express y in terms of x. However, since y is also a function of x, we can see that the equation is not linear in y directly. Instead, we can use an integrating factor approach, but first, we need to express it in a more manageable form.

We can rewrite the equation as:

dy/dx + (tan y)/((1+x)*sec y) = (1+x)e^x.

Finding the Integrating Factor

To solve this, we can look for an integrating factor, which is typically of the form e^(∫P(x)dx). In our case, we need to express the equation in a way that allows us to identify P(x).

However, since the equation involves y in a non-linear way, we might need to consider a substitution or a numerical method if an analytical solution proves too complex. For now, let's assume we can separate variables or use a substitution.

Substitution Approach

Let’s consider a substitution where we let:

u = sec y,

which implies that:

tan y = √(sec²y - 1) = √(u² - 1).

Now, we can express dy in terms of du:

dy = (1/u)(du).

Substituting these into our equation gives us a new form that we can work with. This substitution can simplify the relationship between y and x, allowing us to solve for y more easily.

Solving the Transformed Equation

After substitution, we can rewrite the equation in terms of u and x, leading to a new differential equation that can be solved using standard techniques for first-order equations.

Once we find u(x), we can revert back to y using the inverse of our substitution:

y = sec⁻¹(u).

Final Steps

After solving for u, we can find y explicitly. Depending on the complexity of the resulting equation, we might need numerical methods or further analytical techniques to find an explicit solution.

In summary, while the equation initially appears complex due to the non-linear terms involving y, using substitutions and transforming the equation can lead us to a solution. Each step requires careful manipulation and understanding of the relationships between the variables involved.