Ordinary Differential Equations – JEE Mains Mathematics
1. Introduction to Ordinary Differential Equations (ODEs)
- An ordinary differential equation (ODE) is an equation involving a function and its derivatives with respect to one independent variable.
- It represents the relationship between a function and its rate of change.
- In general, the form of an ODE is: f(x, y, y', y'', ...) = 0, where y is the dependent variable, and x is the independent variable.
2. Order and Degree of Differential Equations
- The **order** of a differential equation is the highest derivative present in the equation.
- The **degree** is the power of the highest derivative when the equation is polynomial in derivatives (i.e., no fractional or negative powers of derivatives).
- Example: For the equation (d²y/dx²) + 2(dy/dx) = 0, the order is 2, and the degree is 1.
3. Solution of Differential Equations
3.1. Solution by the Method of Separation of Variables
- Separation of variables is a method to solve ODEs of the form: f(y)dy = g(x)dx.
- Steps for solving:
- Rearrange the equation to separate variables (y terms on one side and x terms on the other side).
- Integrate both sides of the equation.
- Find the constant of integration using initial conditions (if given).
- Example: Solve the equation (dy/dx) = y/x, which can be rewritten as (1/y)dy = (1/x)dx, and integrate both sides.
3.2. Solution of Homogeneous Differential Equations
- A homogeneous differential equation is one in which the ratio of the dependent and independent variables remains constant.
- The general form of a homogeneous first-order differential equation is:
- (dy/dx) = f(y/x)
- To solve, use substitution: let y = vx, where v = y/x.
- Then, the equation becomes separable, and the solution can be found by integration.
3.3. Solution of Linear Differential Equations
- A linear first-order differential equation has the form:
- (dy/dx) + P(x)y = Q(x)
- To solve, find the integrating factor, I(x), which is given by:
- I(x) = e^(∫P(x) dx)
- Multiply the entire equation by I(x) to make the left-hand side an exact derivative. Then, integrate both sides to find y.
- Example: Solve (dy/dx) + y = e^x. The integrating factor is e^x, and multiplying the equation by e^x gives the solution.
4. General Solution of a Differential Equation
- The general solution of a differential equation contains arbitrary constants, determined by initial conditions or boundary conditions.
- For example, the general solution of (dy/dx) = y/x is y = Cx, where C is the constant of integration.
5. Special Types of Differential Equations
- Exact Differential Equations: An equation of the form M(x, y)dx + N(x, y)dy = 0 is exact if ∂M/∂y = ∂N/∂x. In such cases, we can find the potential function by integrating M(x, y) with respect to x and N(x, y) with respect to y.
- Bernoulli's Differential Equation: A non-linear differential equation of the form (dy/dx) + P(x)y = Q(x)y^n. This can be transformed into a linear equation by using the substitution z = y^(1-n).