Problem 10 - Advanced Engineering Mathematics

Question

\(\frac{d y}{d x}=\left(\frac{2 y+3}{4 x+5}\right)^{2}\)

Step-by-Step Solution

Answer

Separate variables and integrate both sides to find: \( -\frac{1}{2(2y+3)} = -\frac{1}{4(4x+5)} + C \).

Step 1: Identify the Form of the Equation

The given differential equation is \( \frac{d y}{d x} = \left( \frac{2y + 3}{4x + 5} \right)^2 \). This is a Separable Differential Equation, which means we can separate variables on both sides of the equation.

Step 2: Separate the Variables

We rearrange the equation to move all \(y\)-dependent terms to one side and \(x\)-dependent terms to the other: \( \frac{d y}{\left(2y + 3\right)^2} = \frac{d x}{\left(4x + 5\right)^2} \).

Step 3: Integrate Both Sides

Integrate both sides separately: \( \int \frac{d y}{\left(2y + 3\right)^2} = \int \frac{d x}{\left(4x + 5\right)^2} \). Use substitution if necessary: Let \( u = 2y + 3 \) for the left and \( v = 4x + 5 \) for the right.

Step 4: Solve the Left-side Integral

For the left side, let \( u = 2y + 3 \), then \( \frac{1}{2} du = dy \). The integral becomes \( \frac{1}{2} \int \frac{du}{u^2} = -\frac{1}{2u} + C_1 \), where \( C_1 \) is an integration constant.

Step 5: Solve the Right-side Integral

For the right side, let \( v = 4x + 5 \), then \( \frac{1}{4} dv = dx \). The integral becomes \( \frac{1}{4} \int \frac{dv}{v^2} = -\frac{1}{4v} + C_2 \), where \( C_2 \) is another integration constant.

Step 6: Combine and Simplify the Results

Combine the results from both sides: \( -\frac{1}{2(2y+3)} = -\frac{1}{4(4x+5)} + C \), where \( C \) is a constant combining \( C_1 \) and \( C_2 \). Multiply through by the denominators to simplify and solve for \( y \).

Step 7: Solve for y

Re-arrange the expression found in Step 6 to express \( y \) explicitly in terms of \( x \). This step might involve further algebraic manipulation to isolate \( y \).

Key Concepts

Integration

Integration is a fundamental concept used to reverse the process of differentiation. It is represented by the integral sign \( \int \) and is used to find the antiderivative or the area under a curve.
In the context of solving differential equations, integration allows us to find a function given its derivative. For separable differential equations, we integrate each side separately after separating variables.

• The first step in integration is identifying the proper function to integrate.
• Use substitution when necessary to simplify the integral into a basic form.
• Remember that the result of integration includes a constant of integration, such as \( C_1 \) or \( C_2 \), which arises because the indefinite integrals can have several solutions differing by a constant.

This fundamental principle underlies the solution of separable differential equations, wherein the derivative of a function is rewritten in terms of its integrated variables,

Variable Separation

Variable separation is a method used to solve separable differential equations by rearranging the equation so that each side depends on only one variable.
We begin by identifying the given differential equation, in this case, \( \frac{d y}{d x} = \left( \frac{2y + 3}{4x + 5} \right)^2 \), as separable.

• The goal is to manipulate the equation to split all instances involving \( y \) to one side and \( x \) to the other.
• In our equation, this results in \( \frac{dy}{\left(2y + 3\right)^2} = \frac{dx}{\left(4x + 5\right)^2} \).
• This separation enables the integration of the equation as two distinct parts that can be solved individually.

Such separation is crucial because it allows the use of integration techniques directly without additional transformations.

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. Specifically, they describe how a particular quantity changes over time or space.
A separable differential equation, as shown here, is a type of differential equation that can be simplified by separating variables into two different integrable expressions.

• In our example, the rate of change of \( y \) relative to \( x \) is defined by a rational expression involving both \( y \) and \( x \).
• Separable equations can be straightforwardly split by isolating the variables on either side of the equation.
• Solving involves integrating each side of the equation to establish a relationship between \( x \) and \( y \).

This entire process highlights the utility of differential equations in formulating real-world phenomena and finding solutions by determining functional relationships between variables.

Substitution Method

The substitution method is a crucial technique used in calculus for simplifying difficult integrals. It involves changing variables to convert an integral into a more manageable form.
In separable differential equations, substitution can streamline integration by reducing complex expressions into simpler ones.

• Choose a substitution that simplifies the differential expression, such as \( u = 2y + 3 \).
• This choice modifies the integration variable and allows the expression to be easily integrable: \( \frac{1}{2} \int \frac{du}{u^2} \).
• Similarly, on the opposite side, using \( v = 4x + 5 \) simplifies the integration to \( \frac{1}{4} \int \frac{dv}{v^2} \).

After substituting back to the original variables, you express the solutions completely. This substitution is especially helpful where direct integration is too complex or unwieldy.