Problem 22 - Advanced Engineering Mathematics

Question

Let \(M=e^{x}+y\) and \(N=2+x+y e^{y}\) so that \(M_{y}=1=N_{x} . \quad\) From \(f_{x}=e^{x}+y\) we obtain \(f=e^{x}+x y+h(y), h^{\prime}(y)=2+y e^{y},\) and \(h(y)=2 y+y e^{y}-y .\) The solution is \(e^{x}+x y+2 y+y e^{y}-e^{y}=c .\) If \(y(0)=1\) then \(c=3\) and a solution of the initial-value problem is \(e^{x}+x y+2 y+y e^{y}-e^{y}=3\)

Step-by-Step Solution

Answer

Solution is \(e^x + xy + 2y + ye^y - e^y = 3\) for \(y(0)=1\).

Step 1: Understand the Problem

We need to verify the given solution for the differential equation using specified conditions. Functions \( M = e^x + y \) and \( N = 2 + x + y e^y \) need verification of their cross partial derivatives for compatibility.

Step 2: Verify Given Partial Derivatives

We verify if the given partial derivatives are correct: \( M_y = 1 \) and \( N_x = 1 \). Indeed, from \( M = e^x + y \), we have \( M_y = 1 \). From \( N = 2 + x + y e^y \), we have \( N_x = 1 \). Both equalities hold, confirming equality of cross partials.

Step 3: Solve for \( f(x, y) \) Using Integrals

From \( f_x = e^x + y \), integrate with respect to \( x \) to get \( f = e^x + xy + h(y) \). Now, differentiate this with respect to \( y \) to obtain \( f_y = x + h'(y) \).

Step 4: Determine \( h(y) \) Using \( f_y \)

Set \( f_y = M_y = 1 \), giving \( x + h'(y) = 1 \). This step confirms \( h'(y) = 2 + y e^y \) based on provided conditions. Integrate \( h'(y) \) as \( h(y) = 2y + y e^y - y + C' \).

Step 5: Combine Integrations to Find \( f(x, y) \)

Substitute \( h(y) \) back into \( f(x, y) \) to form \( f(x, y) = e^x + xy + 2y + y e^y - e^y + C \).

Step 6: Apply Initial Condition to Calculate Constant

Use \( y(0) = 1 \) with solution \( e^x + xy + 2y + y e^y - e^y = C \). At \( x=0, y=1\), \( f(0, 1) = e^0 + 0 + 2 \cdot 1 + 1 \cdot e^1 - e^1 = 3 \). Thus, \( C = 3 \).

Step 7: Verify the Solution

The differential equation solution considering the initial condition is \( e^x + xy + 2y + y e^y - e^y = 3 \). Confirm that the derivative conditions and compatibility have been ensured, thus the solution is valid.

Key Concepts

Partial Derivatives

Partial derivatives are a critical component in solving and understanding differential equations. When dealing with functions of several variables, like our functions \( M = e^x + y \) and \( N = 2 + x + y e^y \), partial derivatives reflect how the function changes with respect to one variable while keeping others constant.
For verifying a solution involving such functions, we need to check the cross partial derivatives, which should be equal (\( M_y \) should equal \( N_x \)) to ensure compatibility. This property is crucial in validating that a particular function is a potential solution to the differential equation.
In our case, both \( M_y = 1 \) and \( N_x = 1 \), thus confirming that the partial derivatives are compatible. This compatibility helps in simplifying further integrations and finding a function \( f(x, y) \) that satisfies the given differential equation.

Initial Value Problem

An initial value problem in differential equations is effectively a way to find a specific solution that fits the given conditions, typically involving values of the function or its derivatives at a particular point.
In general, the process involves solving the differential equation to obtain a general solution and then using the initial conditions to find specific values for constants in the solution. Here, we are given \( y(0) = 1 \), which helps us determine the constant \( C \) in our derived solution.
This is significant because it transforms a general solution obtained from the differential equation into a solution that uniquely fits the situation described by the initial conditions.

Integration

Integration is a key mathematical operation used to derive functions when given derivatives, as we need to reconstruct the original function, \( f(x, y) \), from its partial derivatives.
In this problem, we start with \( f_x = e^x + y \). Integrating with respect to \( x \) gives us \( f = e^x + xy + h(y) \), where \( h(y) \) is an unknown function of \( y \) that acts as a constant with respect to \( x \).
To determine \( h(y) \), we use further information, such as \( h'(y) = 2 + ye^y \), derived from the condition \( f_y = x + h'(y) = 1 \). We integrate \( h'(y) \) to get \( h(y) \), enabling us to construct the full function \( f(x, y) \).

Solution Verification

Solution verification is an essential step that ensures the proposed solution to a differential equation holds under all specified conditions and constraints.
In this exercise, after forming the function \( f(x, y) \), we verify it against both its partial derivatives and the initial condition given: \( y(0) = 1 \). These checks confirm whether the function satisfies the equation under these conditions.
The final solution obtained, \( e^x + xy + 2y + ye^y - e^y = 3 \) when the initial condition is applied, ensures that not only does the equation work for \( x \) and \( y \) individually, but also at the specific point given by the initial condition. Thus, verifying such equations assures their usage and application are mathematically sound.