Exercise 9.3 Class 12th Math

Exercise 9.3 Class 12th Math: A full breakdown to Differential Equations

This article provides a detailed walkthrough of Exercise 9.3 from the Class 12th mathematics textbook, focusing on differential equations. Consider this: we'll cover each problem, providing step-by-step solutions and explanations to enhance your understanding of the concepts involved. This guide aims to be more than just an answer key; it's designed to help you master the techniques necessary to solve a wide range of differential equations.

Introduction to Differential Equations

Before we dive into the exercises, let's refresh our understanding of differential equations. Think about it: a differential equation is an equation that involves a function and its derivatives. The order of a differential equation is the order of the highest derivative present in the equation. To give you an idea, dy/dx + y = x is a first-order differential equation, while d²y/dx² + dy/dx + y = 0 is a second-order differential equation. This exercise focuses primarily on first-order differential equations, which can be solved using various methods.

Types of First-Order Differential Equations Covered in Exercise 9.3

Exercise 9.3 typically covers several types of first-order differential equations, including:

• Variable Separable Equations: These equations can be written in the form f(x)dx = g(y)dy, allowing for direct integration to find the solution.
• Homogeneous Differential Equations: These equations are of the form dy/dx = f(x, y), where f(x, y) is a homogeneous function of degree zero (i.e., f(λx, λy) = f(x, y) for any scalar λ). These are solved by substituting y = vx and separating variables.
• Linear Differential Equations: These are equations of the form dy/dx + Py = Q, where P and Q are functions of x. These are solved using an integrating factor.
• Exact Differential Equations: These equations are of the form M(x, y)dx + N(x, y)dy = 0, where ∂M/∂y = ∂N/∂x. The solution is found by integrating a potential function.

Detailed Walkthrough of Exercise 9.3 Problems (Example Problems)

Since I don't have access to a specific textbook, I will present example problems representing the types of questions typically found in Exercise 9.3 of a Class 12th mathematics textbook focusing on differential equations. Remember to refer to your textbook for the exact problems.

Example Problem 1: Variable Separable Equation

Solve the differential equation: dy/dx = e^(x+y)

Solution:

1. Rewrite the equation: We can rewrite the equation as dy/dx = e^x * e^y.
2. Separate variables: Divide both sides by e^y and multiply by dx: e^(-y)dy = e^x dx.
3. Integrate both sides: ∫e^(-y)dy = ∫e^x dx
4. Solve the integrals: -e^(-y) = e^x + C (where C is the constant of integration)
5. Solve for y: e^(-y) = -e^x - C => -y = ln(-e^x - C) => y = -ln(-e^x - C)

Example Problem 2: Homogeneous Differential Equation

Solve the differential equation: dy/dx = (x² + y²)/(xy)

Solution:

1. Check for homogeneity: The function (x² + y²)/(xy) is homogeneous of degree 0 because if we replace x with λx and y with λy, we get (λ²x² + λ²y²)/(λxλy) = (x² + y²)/(xy).
2. Substitute y = vx: dy/dx = v + x(dv/dx)
3. Substitute into the equation: v + x(dv/dx) = (x² + v²x²)/(vx²) = (1 + v²)/v
4. Separate variables: x(dv/dx) = (1 + v²)/v - v = 1/v
5. Integrate both sides: ∫v dv = ∫dx/x
6. Solve the integrals: v²/2 = ln|x| + C
7. Substitute back y = vx: y²/2x² = ln|x| + C
8. Solve for y: y² = 2x²(ln|x| + C)

Example Problem 3: Linear Differential Equation

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Solve the differential equation: dy/dx + y = x

Solution:

1. Identify P and Q: P = 1 and Q = x
2. Find the integrating factor: I.F. = e^(∫P dx) = e^(∫1 dx) = e^x
3. Multiply the equation by the integrating factor: e^x(dy/dx) + ye^x = xe^x
4. Rewrite the left side as a derivative: d/dx(ye^x) = xe^x
5. Integrate both sides: ∫d/dx(ye^x) dx = ∫xe^x dx
6. Solve the integral on the right side (using integration by parts): ∫xe^x dx = xe^x - e^x + C' (where C' is the constant of integration)
7. Solve for y: ye^x = xe^x - e^x + C' => y = x - 1 + C'e^(-x)

Example Problem 4: Exact Differential Equation

Solve the differential equation: (2xy + y²)dx + (x² + 2xy)dy = 0

Solution:

1. Check for exactness: M(x, y) = 2xy + y² and N(x, y) = x² + 2xy. ∂M/∂y = 2x + 2y and ∂N/∂x = 2x + 2y. Since ∂M/∂y = ∂N/∂x, the equation is exact.
2. Find the potential function: ∫M dx = ∫(2xy + y²)dx = x²y + xy² + h(y)
3. Differentiate with respect to y: ∂/∂y(x²y + xy² + h(y)) = x² + 2xy + h'(y)
4. Compare with N(x, y): x² + 2xy + h'(y) = x² + 2xy => h'(y) = 0
5. Solve for h(y): h(y) = C'' (where C'' is a constant)
6. Write the solution: x²y + xy² + C'' = C (where C is the constant of integration) => x²y + xy² = C - C'' = K (where K is a constant)

Frequently Asked Questions (FAQ)

• What if I get stuck on an integral? Practice is key! Review integration techniques, including substitution, integration by parts, and partial fractions. If you're still stuck, consult your textbook or seek help from a teacher or tutor.
• How do I know which method to use? The form of the differential equation will guide your choice of method. Look for patterns that indicate variable separable, homogeneous, linear, or exact forms.
• What are some common mistakes to avoid? Common mistakes include errors in integration, incorrect separation of variables, and forgetting the constant of integration. Carefully check each step of your work.
• How can I improve my understanding of differential equations? Practice solving a variety of problems, review the theory thoroughly, and don't hesitate to seek help when needed.

Conclusion

Mastering differential equations requires consistent effort and practice. This guide provides a foundation for understanding the various methods involved in solving first-order differential equations, often encountered in Exercise 9.And 3 of Class 12th mathematics textbooks. Remember to work through the problems in your textbook, applying the techniques discussed here. In practice, focus on understanding the underlying principles, not just memorizing solutions. With dedication and practice, you will develop the skills needed to confidently tackle these and more complex differential equations. Remember to always check your answers and compare them against your textbook's solutions to identify any potential areas where your understanding needs further refinement. Good luck!