Class 12 Maths Chapter 9 Differential Equations Notes

Key Definitions

Differential Equation

An equation involving an independent variable, a dependent variable, and derivatives of the dependent variable with respect to the independent variable.

Order of a Differential Equation

The order of the highest order derivative appearing in the differential equation.

Degree of a Differential Equation

The highest power of the highest order derivative in a differential equation, when it is expressed as a polynomial in derivatives. It must be defined for the equation to be a polynomial in its derivatives.

General Solution

A solution to a differential equation that contains as many arbitrary constants as the order of the differential equation. It represents a family of curves.

Particular Solution

A solution obtained from the general solution by assigning specific values to the arbitrary constants, usually determined by initial or boundary conditions. It represents a unique curve.

Homogeneous Differential Equation

A first-order, first-degree differential equation of the form dy/dx = f(x,y) or dx/dy = g(x,y), where f(x,y) and g(x,y) are homogeneous functions of degree zero.

Linear Differential Equation

A first-order differential equation that can be written in the form dy/dx + Py = Q or dx/dy + Px = Q, where P and Q are functions of x (or constants) or functions of y (or constants), respectively.

Understanding Differential Equations and Their Solutions

Differential equations are fundamental tools in mathematics and science, used to model dynamic systems where quantities change. For instance, population growth, radioactive decay, and circuit analysis are all described by differential equations. A differential equation establishes a relationship between a function and its derivatives, providing insight into how a system evolves over time or space.

The first crucial step in dealing with any differential equation is to correctly identify its order and degree. The order is straightforward: it's simply the highest derivative present (e.g., d²y/dx² means order 2). The degree is slightly more complex; it's the power of that highest derivative after the equation has been cleared of radicals and fractions in terms of derivatives. If an equation cannot be expressed as a polynomial in its derivatives (e.g., sin(dy/dx)), its degree is undefined.

Once classified, solving a differential equation involves finding the function y(x) (or x(y)) that satisfies the equation. The solution derived will typically be a general solution, containing arbitrary constants. These constants signify a family of curves. For example, y = Ce^x is the general solution to dy/dx = y. Each value of C gives a different curve. When specific initial conditions (e.g., y(0) = 1) are provided, we can determine the exact value of these constants, leading to a particular solution, which represents a unique curve from the family. This process of finding a particular solution from a general one by applying given conditions is a very common type of problem in board exams. Mastering the various techniques—variables separable, homogeneous equations, and linear differential equations—is key to success in this chapter.

Key Formulas and Solution Strategies

• Order & Degree: Always ensure the DE is a polynomial in derivatives before finding the degree. Otherwise, the degree is undefined.
• Formation of DE: To form a differential equation, differentiate the given family of curves repeatedly until all arbitrary constants are eliminated.
• Variables Separable: If dy/dx = f(x)g(y), separate variables to get dy/g(y) = f(x)dx and then integrate both sides.
• Homogeneous DE: For dy/dx = f(x,y) where f(x,y) is homogeneous of degree zero, substitute y = vx (so dy/dx = v + x dv/dx) to convert it into a variables separable form.
• Linear Differential Equation (Type 1): dy/dx + Py = Q. Integrating Factor (IF) = e^(∫Pdx). Solution: y.(IF) = ∫Q.(IF)dx + C.
• Linear Differential Equation (Type 2): dx/dy + Px = Q. Integrating Factor (IF) = e^(∫Pdy). Solution: x.(IF) = ∫Q.(IF)dy + C.
• General Solution vs. Particular Solution: General solution has arbitrary constants; particular solution is found using given initial/boundary conditions.
• Checking Solutions: To verify if a function is a solution, substitute the function and its derivatives back into the original differential equation.

Steps to Solve a Linear Differential Equation

1. Step 1: Identify the Form — Check if the equation is in the standard form dy/dx + Py = Q or dx/dy + Px = Q. P and Q must be functions of x (or constants) for the first form, and functions of y (or constants) for the second.
2. Step 2: Find the Integrating Factor (IF) — For dy/dx + Py = Q, the IF is e^(∫Pdx). For dx/dy + Px = Q, the IF is e^(∫Pdy). Remember the integral is with respect to the independent variable in the denominator.
3. Step 3: Apply the Solution Formula — If dy/dx + Py = Q, the general solution is y (IF) = ∫ Q (IF) dx + C. If dx/dy + Px = Q, the general solution is x (IF) = ∫ Q (IF) dy + C.
4. Step 4: Integrate and Simplify — Perform the integration on the right-hand side. Don't forget the constant of integration C. Simplify the expression to get the general solution.
5. Step 5: Find Particular Solution (if required) — If initial or boundary conditions are given (e.g., y(0)=1), substitute these values into the general solution to find the value of C and thus the particular solution.

Worked Examples

• {"title":"Example 1: Variables Separable","description":"Q: Solve dy/dx = (1+y²)/(1+x²). \nA: Separate variables: dy/(1+y²) = dx/(1+x²). \nIntegrate both sides: ∫dy/(1+y²) = ∫dx/(1+x²). \nResult: tan⁻¹y = tan⁻¹x + C."}
• {"title":"Example 2: Linear Differential Equation","description":"Q: Solve dy/dx + (y/x) = x². \nA: This is in the form dy/dx + Py = Q, where P = 1/x and Q = x². \nIF = e^(∫(1/x)dx) = e^(ln|x|) = x. \nSolution: y x = ∫ x² x dx + C \nxy = ∫ x³ dx + C \nxy = x⁴/4 + C."}

Exam Tip: Avoiding Common Traps

When solving differential equations, students often lose marks due to algebraic errors or forgetting the constant of integration. Always check the form of the equation carefully before deciding on the solution method. If an equation doesn't immediately look like variables separable or linear, check if it's homogeneous or can be reduced to one of these forms. For linear differential equations, ensure P and Q are correctly identified and don't forget the absolute value in ln|x| when integrating 1/x for the integrating factor, though e^(ln|x|) often simplifies to x. Double-check the integration steps, especially for ∫Q.(IF)dx. For particular solutions, substitute the given conditions carefully to find C.

Practice Questions with Solutions

• Q: What is the order and degree of the differential equation (d²y/dx²)³ + (dy/dx)² + sin(dy/dx) = 0? A: Order = 2, Degree = Not Defined (due to sin(dy/dx)).
• Q: State the general form of a linear differential equation of the first order where the independent variable is y. A: dx/dy + Px = Q, where P and Q are functions of y or constants.
• Q: What substitution is typically used to solve a homogeneous differential equation of the form dy/dx = f(y/x)? A: Substitute y = vx, which implies dy/dx = v + x(dv/dx).
• Q: If IF = e^(∫Pdx), what is the general solution for dy/dx + Py = Q? A: y (IF) = ∫ Q (IF) dx + C.

Frequently Asked Questions