Problem 4 - Calculus 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Various standard and special methods Applied to different type of the differential equation of the first order and first degree arc as follows: (i) Method of seperation of variables: \(f(x)\) \(d x=g(y) d y\) (ii) Reduciable to variable scparable from: \(\frac{d y}{d x}=f(a x+b y+c)\) Method of substitution: pul \(a x+b y+\) \(c=V\) (iii) Homogeneous differential equation \(\frac{d y}{d x}=\frac{f_{1}(x, y)}{f_{2}(x, y)}=\frac{x^{n} f_{1}(y / x)}{x^{n} f_{2}(y / x)}\) where, \(f_{1}, f_{2}\) are the polynomial in \(x, y\) put \(\frac{y}{x}=v\) or \(y \quad v x\) sothat \(\left\\{\frac{d y}{d x}=v+x \frac{d v}{d x}\right\\}\) with these two substitution homogenious diflerential cquations reduce to variable separable form. (iv) Non-homogeneous differential equation: \(\frac{d y}{d x}=\frac{a x+b y+c}{a^{\prime} x+b^{\prime} y+c}\)

Step-by-Step Solution

Answer

Use separation of variables, appropriate substitutions, and simplifications to solve first-order differential equations.

Step 1: Understanding the method of Separation of Variables

The differential equation takes the form \( f(x) \, dx = g(y) \, dy \). This means the problem can be approached by separating the variables related to \( x \) and \( y \) on either side of the equation. Once the variables are separated, integrate both sides to solve for \( y \).

Step 2: Applying Separation of Variables

Reorganize the equation into \( \int f(x) \, dx = \int g(y) \, dy \). Integrate both sides: the left side with respect to \( x \) and the right side with respect to \( y \), providing the solution implicitly if an explicit formula for \( y \) is difficult to determine.

Step 3: Reduction to Variables Separable Form

For the equation \( \frac{dy}{dx} = f(ax + by + c) \), use the substitution \( ax + by + c = V \), implying \( dV = (a \frac{dx}{dx} + b \frac{dy}{dx}) \, dx \). This reduces the equation to a separable form in terms of \( V \) and either \( x \) or \( y \). Follow the separation of variables method to solve.

Step 4: Solving Homogeneous Differential Equations

A homogeneous differential equation has the form \( \frac{dy}{dx} = \frac{f_1(x, y)}{f_2(x, y)} \) where the functions \( f_1 \) and \( f_2 \) are polynomials of the same degree. Use the substitution \( y = vx \) (where \( v \) is a function of \( x \)), leading to \( \frac{dy}{dx} = v + x \frac{dv}{dx} \). Substitute and simplify to find \( dv \) and \( dx \) isolations, which allows separation and integration.

Step 5: Solving Non-Homogeneous Differential Equations

For equations of the form \( \frac{dy}{dx} = \frac{ax + by + c}{a'x + b'y + c'} \), attempt to simplify by inspecting specific substitutions or transformations. An integrating factor or a change of variables may assist in solving it, depending on the simplification possible. If an exact solution isn't easy, consider numerical solutions or approximations.

Key Concepts

Separation of Variables

The method of Separation of Variables is a staple technique for solving first order differential equations. When you come across an equation in the form of \( f(x) \, dx = g(y) \, dy \), it's the ideal method to apply. To use this method:

• Isolate the variables on different sides of the equation, which means having all the \( x \)-related terms on one side and \( y \)-related terms on the other.
• Once separated, integrate both sides. You'll integrate \( f(x) \) with respect to \( x \) and \( g(y) \) with respect to \( y \).

This results in a relationship between \( x \) and \( y \) that may not always be easy to solve explicitly for \( y \), but it still provides valuable insights into the solution’s behavior. If finding an explicit solution isn't straightforward, leave the result in its implicit form.

Homogeneous Equations

Homogeneous Equations hold a unique characteristic: they maintain proportional structure. In more technical terms, an equation is homogeneous if it satisfies \( \frac{dy}{dx} = \frac{f_1(x, y)}{f_2(x, y)} \), with both \( f_1 \) and \( f_2 \) being polynomials of the same degree. Solving these involves the clever substitution:

• Start with the substitution \( y = vx \), where \( v \) is a function of \( x \).
• This transforms the equation into one that can be handled similarly to a separable equation, as \( \frac{dy}{dx} = v + x \frac{dv}{dx} \).
• Reorganize the terms to isolate \( v \) and \( x \), making it easier to integrate.

This substitution exploits the homogeneous nature, allowing a transformation that simplifies the problem to a more manageable form.

Non-Homogeneous Equations

Non-Homogeneous Equations are similar to their homogeneous counterparts, but with an added twist: they include terms that prevent the equation from maintaining the same degree across all terms. Solving such an equation, especially one like \( \frac{dy}{dx} = \frac{ax + by + c}{a'x + b'y + c'} \), often requires additional techniques:

• Check if a substitution helps, such as reducing the equation into a form that resembles a homogeneous equation or another form that's easier to solve.
• If straightforward simplifications aren't apparent, look for integrating factors or transformations.

Integrating factors and transformations can facilitate the process by changing the equation's structure, making the solution more tractable. When an analytic solution is elusive, numerical methods or approximations might provide a practical alternative.

Variable Substitution

Variable Substitution is a powerful strategy in the toolbox of differential equations, especially useful when facing equations that are stubbornly complex in their original form. Consider an equation \( \frac{dy}{dx} = f(ax + by + c) \). To simplify:

• Make a substitution, such as \( ax + by + c = V \), which effectively reduces the complexity of the original problem.
• After substituting, the differentiation shifts, altering \( dy/dx \) into a separable form relative to \( V \) and either \( x \) or \( y \).

Once simplified, tackle the problem using the separation of variables method. By strategically choosing substitutions, you transform an intimidating equation into one that's approachable, highlighting the power of strategic mathematical maneuvers.