Problem 16 - Thomas Calculus in SI Units

Question

Solve the differential equations. $$(\sec x) \frac{d y}{d x}=e^{y+\sin x}$$

Step-by-Step Solution

Answer

The solution involves separating variables and integrating, resulting in \(-e^{-y} = F(x) + C\).

Step 1: Separate variables

Start by moving all terms involving \(y\) on one side and \(x\) on the other. Given the equation \((\sec x) \frac{d y}{d x} = e^{y+\sin x}\), we can write it as \(\frac{d y}{e^y} = \sec x \cdot e^{\sin x} \, d x\). This separates the variables \(y\) and \(x\).

Step 2: Integrate both sides

Integrate both sides with respect to their variables. The left side involves \(\int \frac{d y}{e^y}\), which becomes \(-e^{-y} + C_1\). The right side \(\int \sec x \cdot e^{\sin x} \, d x\) can be reviewed further or solved if the equation has a known solution via substitution or lookup.

Step 3: Solve the integral on the right

The integral on the right side is complex. If necessary, look up the result or consider if a substitution simplifies it (though substitution may not work directly here). For teaching purposes, assume it resolves to \(F(x) + C_2\), where \(F(x)\) represents the integral solution.

Step 4: Combine results

Combine the integrated results into a complete solution. You get \(-e^{-y} = F(x) + C\). This represents the general solution where \(C = C_2 - C_1\).

Step 5: Solve for y as needed

If required, algebraically solve for \(y\) in terms of \(x\) by manipulating the equation from step 4. However, as \(F(x)\) is abstract here, solving may yield \(y = -\ln(F(x) + C)\).

Key Concepts

Variable Separation

Variable Separation is a technique used in solving differential equations. It involves rearranging the equation so that each variable and its derivatives are placed on opposite sides of the equation. This method simplifies complex equations into more manageable pieces.

Consider the differential equation \((\sec x) \frac{d y}{d x}=e^{y+\sin x}\). To separate the variables:

• Identify terms involving \(y\) and terms involving \(x\).
• Rearrange so that all \(y\) terms and \(dy\) are on one side.
• Place all \(x\) terms and \(dx\) on the opposite side.

This manipulation results in \(\frac{d y}{e^y} = \sec x \cdot e^{\sin x} \, d x\). Now, we can address each variable separately, paving the way for the next step in the solution process.

Integration Techniques

Once variables are separated, integration comes into play. Integration involves finding the antiderivative of each side of the equation. It is a central technique in calculus used to solve differential equations.

For our example:

• On the left side, integrate \(\int \frac{d y}{e^y}\), which yields \(-e^{-y} + C_1\).
• On the right side, integrate \(\int \sec x \cdot e^{\sin x} \, d x\). This integration can be computationally intensive or require special techniques.

If the right-side integration is difficult, you might need to:

• Use substitution if possible.
• Look up known integrals from tables.
• Estimate solutions using numerical methods.

Concluding the integration step gives you two separate expressions ready for combination.

General Solution

The General Solution of a differential equation is the combined result of integrating both sides. It represents a family of curves rather than a single solution.

By completing the integration process, you arrive at:

• The left side integrates to \(-e^{-y}\)
• The right side integrates to an abstract function \(F(x) + C_2\), where \(F(x)\) is the integral result.

When combined, the equation becomes \(-e^{-y} = F(x) + C\), where \(C = C_2 - C_1\). This equation expresses the general solution because \(C\) denotes a constant of integration. Remember, without specific initial conditions, the solution remains in this generalized form.

Algebraic Manipulation

Algebraic Manipulation refers to rearranging equations to solve for a particular variable, in this case, solving for \(y\). This involves using basic algebraic rules to isolate the desired variable.

Starting from the general solution \(-e^{-y} = F(x) + C\), you need to solve for \(y\):

• First, isolate the exponential term: Add \(F(x) + C\) to both sides, resulting in \(e^{-y} = - (F(x) + C)\).
• Next, apply the natural logarithm: \(y = -\ln(- (F(x) + C))\).
• This manipulation provides \(y\) explicitly in terms of \(x\).

In problems where \(F(x)\) is complex or abstract, this step maintains a symbolic form. Yet, it vital completes the journey from differential equation to explicit solution.