Problem 44 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Solution of the equation \(\frac{d y}{d x}=e^{x-y}\left(e^{x}-e^{y}\right)\) is (A) \(e^{y}=e^{x}-1+c e^{-e^{\prime}}\) (B) \(e^{y}=e^{x}-1+c e^{e^{*}}\) (C) \(e^{x}=e^{y}-1+c e^{-e^{\prime}}\) (D) None of these

Step-by-Step Solution

Answer

The differential equation solution isn't directly solvable into given forms; hence (D) None of these is plausible.

Step 1: Identify the Differential Equation Type

The given differential equation is \( \frac{d y}{d x}=e^{x-y}(e^{x}-e^{y}) \). This is a first-order differential equation and looks separable as we can express the terms involving \( y \) on one side and \( x \) on the other.

Step 2: Separate Variables

Rewrite the equation to separate the variables. Start by expressing as:\[ \frac{d y}{d x} = e^{x}e^{-y} \cdot (e^x - e^y) \]This simplifies the equation to try and get all \( y \)-terms on one side and \( x \)-terms on the other.

Step 3: Simplify and Integrate

Attempt to express it in the form \( M(x)dx + N(y)dy = 0 \).From the separation: \( e^{y} dy = e^xe^{-y} (e^x - e^y) dx \), simplify to integral terms. Though tricky, initial approach would use algebraic manipulation to bring into standard form before integrating.

Step 4: Verify Separability with Incorrect Attempts

With complex forms, errors suggest misinterpretation. Checking option offers insight:Using potential identity can caution off confirming options such as if \( e^y = e^x - 1 + ce^{-e'} \) satisfy base rearranged states substituting back fully aligns with study enjoy!

Key Concepts

Variable Separable Method

In solving first-order differential equations, one effective strategy is the variable separable method. This technique is most useful when we can express the equation in a way that the variables can stand alone on each side of the equation.

The main idea is to rearrange the equation such that:

• All terms involving the dependent variable (e.g., \( y \)) are on one side
• All terms involving the independent variable (e.g., \( x \)) are on the other side

Step by step, we identify this possibility by inspecting if the given equation logically allows for such separation. For instance, in the original exercise, observe the expression: \( \frac{dy}{dx} = e^{x-y}(e^{x} - e^{y}) \). Recognizing it as separable means acknowledging our ability to independently manage \( x \) and \( y \) within their own respective realms.

Once separation is complete, the equation turns into an ideal candidate for integration, thus transitioning us to the next fundamental step.

Integration

After successfully separating the variables, the next crucial step is integration. This mathematical process allows us to solve for the general solution of the differential equation.

In its essence, integration is the operation of finding the integral of a function—a reverse of differentiation. Once the variables are separated, integrating each side with respect to their respective variable is essential.

• The left side of the equation, concerning the dependent variable, will be integrated with respect to \( y \)
• The right side, involving the independent variable, will be integrated with respect to \( x \)

Consider the integrals:\[\int e^{y} dy \]\[\int e^x e^{-y}(e^x-e^y) dx\]Breaking it down and performing these integrations will lead to an expression linking \( x \) and \( y \) through a constant of integration. This reveals the solution curve on which the differential equation's behavior is based.

Algebraic Manipulation

Even after separating variables and integrating, algebraic manipulation remains key in simplifying and solving the equation.

Once the integrations are done, it might result in a complicated expression, which requires careful handling to isolate the desired variable. This involves simplifying expressions, performing algebraic operations, and sometimes assuming initial conditions or known values to find the constant of integration.

• Rewriting expressions to achieve a clear form
• Applying identities or rearrangements to verify the solution
• Utilizing known functions or properties to further simplify results

Returning to our example equation, manipulation helps in revisiting obtained results to scrutinize solution options, like evaluating logical derivatives or ensuring form adherence. This discipline confirms whether the solution satisfies all given conditions, allowing for effective understanding and application of the solved differential equation.