Differential Equation.pptx free download

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1. Differential Equation Name: Ahmad Anwar Hassan Shah

2. Linear First Order A linear first-order differential equation is a differential equation of the form: y is the unknown function x is the independent variable P(x) and Q(x) are functions of x dy/dx is the derivative of y with respect to x

3. The general solution of the differential equation is: y = (1/e^∫P(x)dx) ∫(Q(x)e^∫P(x)dx)dx + Ce^(-∫P(x)dx) where C is the constant of integration. Here are some examples:

4. Solve the differential equation: dy/dx + 2y = 3 Solution: Integrating factor = e^∫2dx = e^(2x) Multiply both sides by the integrating factor: e^(2x)dy/dx + 2e^(2x)y = 3e^(2x) Now, integrate both sides: ∫(e^(2x)dy) + ∫(2e^(2x)y)dx = ∫(3e^(2x))dx e^(2x)y = (3/2)e^(2x) + C y = (3/2) + Ce^(-2x)

5. Solve the differential equation: dy/dx - 3y = 2 Solution: Integrating factor = e^∫(-3)dx = e^(-3x) Multiply both sides by the integrating factor: e^(-3x)dy/dx - 3e^(-3x)y = 2e^(-3x) Now, integrate both sides: ∫(e^(-3x)dy) - ∫(3e^(-3x)y)dx = ∫(2e^(-3x))dx e^(-3x)y = (-2/3)e^(-3x) + C y = (-2/3) + Ce^(3x)

6. Numerical Problems Solve the differential equation: dy/dx + 4y = 5, with initial condition y(0) = 1 Solution: Integrating factor = e^∫4dx = e^(4x) Multiply both sides by the integrating factor: e^(4x)dy/dx + 4e^(4x)y = 5e^(4x) Now, integrate both sides: ∫(e^(4x)dy) + ∫(4e^(4x)y)dx = ∫(5e^(4x))dx e^(4x)y = (5/4)e^(4x) + C Using the initial condition y(0) = 1, we get: 1 = (5/4) + C C = -1/4 y = (5/4)e^(-4x) + (1/4)e^(-4x) y = (1/4)(5e^(-4x) + 1)

7. Solve the differential equation: dy/dx - 2y = 3x, with initial condition y(1) = 2 Solution: Integrating factor = e^∫(-2)dx = e^(-2x) Multiply both sides by the integrating factor: e^(-2x)dy/dx - 2e^(-2x)y = 3xe^(-2x) Now, integrate both sides: ∫(e^(-2x)dy) - ∫(2e^(-2x)y)dx = ∫(3xe^(-2x))dx e^(-2x)y = (-3/2)xe^(-2x) + (3/2)e^(-2x) + C Using the initial condition y(1) = 2, we get: 2 = (-3/2)e^(-2) + (3/2)e^(-2) + C C = 2e^2 y = (-3/2)xe^(-2x) + (3/2)e^(-2x) + 2e^(2-2x) y = (-3/2)xe^(-2x) + (1/2)e^(-2x) + 2e^(-2x) Note: e^(2-2x) = e^(2)e^(-2x) = e^2e^(-2x)