Integral Calculus - Solution to differential equation and sign of derivative
- In this lecture, we will discuss the solution to differential equations and how to determine the sign of the derivative.
Differential Equations
- A differential equation relates a function and its derivatives.
- It can be expressed in the form:
dy/dx = f(x)
Where y is the dependent variable and f(x) is a given function.
General Solution
- The general solution of a differential equation contains one or more constants that can take any value.
- It is obtained by integrating both sides of the equation.
- The general solution can be expressed as:
y = F(x) + C
Where F(x) is the antiderivative of f(x) and C is the constant of integration.
Particular Solution
- A particular solution is obtained by assigning specific values to the constants in the general solution.
- These values are determined by the boundary conditions or initial conditions given in the problem.
- The particular solution satisfies both the differential equation and the given conditions.
Example 1
- Find the general solution of the differential equation:
dy/dx = 2x
- Integrating both sides with respect to x, we get:
∫dy = ∫2xdx
y = x^2 + C
- So, the general solution is y = x^2 + C.
Example 2
- Find the particular solution of the differential equation:
dy/dx = 2x
- Given the condition y(0) = 3, we can use it to find the value of C.
- Substituting x=0 and y=3 into the general solution, we get:
3 = 0^2 + C
C = 3
- So, the particular solution is y = x^2 + 3.
Sign of the Derivative
- The sign of the derivative of a function indicates whether the function is increasing or decreasing.
- If the derivative is positive, the function is increasing.
- If the derivative is negative, the function is decreasing.
Example 1
- Consider the function f(x) = x^2.
- To determine the sign of the derivative, we take the derivative f’(x):
f’(x) = 2x
- For x < 0, the derivative is negative, indicating that the function is decreasing.
- For x > 0, the derivative is positive, indicating that the function is increasing.
Example 2
- Consider the function g(x) = 2x - 1.
- To determine the sign of the derivative, we take the derivative g’(x):
g’(x) = 2
- The derivative is a constant 2, which is positive for all values of x.
- This means that the function is always increasing.
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