Problem 27 - Advanced Engineering Mathematics

Question

Solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=2+\sqrt{y-2 x+3} $$

Step-by-Step Solution

Answer

The general solution is \( y = \frac{(x + C)^2}{4} + 2x - 3 \).

Step 1: Identify Suitable Substitution

The right-hand side of the differential equation, \(2 + \sqrt{y - 2x + 3}\), suggests using a substitution that simplifies the square root expression. Let \( v = y - 2x + 3 \), then \( y = v + 2x - 3 \). This substitution will help us simplify the equation.

Step 2: Express the Derivative with Respect to the Substitution

Differentiate \( y = v + 2x - 3 \) with respect to \( x \) to find \( \frac{d y}{d x} \) in terms of \( v \). We get: \( \frac{d y}{d x} = \frac{d v}{d x} + 2 \). Substituting this into the original equation gives: \( 2 + \frac{d v}{d x} = 2 + \sqrt{v} \).

Step 3: Simplify and Solve for the Derivative

By cancelling 2 on both sides, the equation reduces to \( \frac{d v}{d x} = \sqrt{v} \). This is a separable differential equation.

Step 4: Separate and Integrate

Separate the variables \( v \) and \( x \): \( \frac{d v}{\sqrt{v}} = dx \). Integrate both sides: \( \int \frac{d v}{\sqrt{v}} = \int dx \). The integral on the left is \( 2\sqrt{v} \) and on the right is \( x + C \), where \( C \) is the constant of integration.

Step 5: Solve for v

Equating the integrals, we get: \( 2\sqrt{v} = x + C \). Solve for \( v \) by squaring both sides: \( v = \frac{(x + C)^2}{4} \).

Step 6: Back Substitute to Solve for y

Recall the substitution \( v = y - 2x + 3 \). Substitute back to find \( y \): \( y - 2x + 3 = \frac{(x + C)^2}{4} \). Thus, \( y = \frac{(x + C)^2}{4} + 2x - 3 \).

Step 7: Present the General Solution

The general solution to the differential equation is \( y = \frac{(x + C)^2}{4} + 2x - 3 \), where \( C \) is an arbitrary constant.

Key Concepts

Separable Differential Equations

A separable differential equation is one that can be manipulated so that all terms containing the dependent variable are on one side of the equation and all terms containing the independent variable are on the other. This allows us to integrate each side with respect to its variable, effectively separating the equation. In our exercise, after substitution and simplification, the equation \( \frac{d v}{d x} = \sqrt{v} \) is separable. We rearrange it to \( \frac{d v}{\sqrt{v}} = dx \), making it easy to integrate both sides separately.

• The key is recognizing the form: \( f(v) \) on one side and \( g(x) \) on the other.
• Usually, separability is a straightforward process once the equation is correctly simplified.

Understanding how to manipulate a differential equation to be separable is essential for solving it.

Substitution Method

The substitution method is a powerful technique used to simplify differential equations by introducing a new variable. In our example, the equation initially seemed complex due to the square root in \( 2 + \sqrt{y - 2x + 3} \). By setting \( v = y - 2x + 3 \), we transformed the problem into a form that's easier to handle.

• This new variable \( v \) simplifies the square root, making integration more manageable.
• After the substitution, the complex expression becomes a simpler differential equation \( 2 + \frac{dv}{dx} = 2 + \sqrt{v} \).

Substitution is often used when particular components of an equation complicate direct integration, allowing us to work with more straightforward expressions.

Integration Techniques

Integration is a critical part of solving differential equations, especially when dealing with expressions like \( \int \frac{d v}{\sqrt{v}} \). These techniques include:

• Basic Integration: Handle standard integrable forms, such as \( \int dx = x + C \).
• Integration by Substitution: When substitution simplifies the integration process, as \( \frac{d v}{\sqrt{v}} \) yields \( 2\sqrt{v} \).

The integration step often leads directly to solving for the variable of interest. In our problem, once the equation is separated, integration becomes straightforward, leading to \( 2\sqrt{v} = x + C \). There, the constant of integration \( C \) plays a vital role in representing the general solution's family of curves.

General Solution

The general solution of a differential equation encompasses all possible solutions, typically described in terms of arbitrary constants. For the given problem, the general solution is \( y = \frac{(x + C)^2}{4} + 2x - 3 \). This solution arises from solving for \( y \) after completing all integration and substitution steps.

• The arbitrary constant \( C \) represents different specific solutions depending on initial conditions or additional constraints.
• The form of the general solution often tells us about the behavior of the system or function described by the differential equation.

Knowing how to derive and interpret the general solution is crucial, as it reflects the full range of possible solutions for the initial differential equation under various conditions.