Minor point, but this should be f(0) = 0, f'(0) = 1.chapsticks said:Homework Statement
Solve the following differential equation:
f"(x)=sinx
x(0)=0, x'(0)=1
What's your question?chapsticks said:Homework Equations
x(0)=0, x'(0)=1
The Attempt at a Solution
f"(x)=sin(x)
integrate,
f'(x)=-cos(x)+C1
f'(0)=-cos(0)+C1=1 => C1=2
C1=2
f'(x)=-cos(x)+2
f(x)=-sin(x)+2x+C2
f(0)=-sin(0)+C2=0 => C2=0
=>
f(x)=-sin(x)+2x
A differential equation is a mathematical equation that relates a function to its derivatives. It describes how a quantity changes over time and can be used to model a wide range of phenomena in physics, engineering, and other scientific fields.
Differential equations are important because they allow us to mathematically model and understand complex systems and processes. They are used in many areas of science and engineering, such as predicting the motion of planets, designing electrical circuits, and studying population dynamics.
There are several methods for solving differential equations, including separation of variables, integrating factors, and numerical methods. The specific method used depends on the type of differential equation and the variables involved.
Ordinary differential equations involve a single independent variable, such as time, and one or more dependent variables. Partial differential equations involve multiple independent variables and partial derivatives of the dependent variables. They are typically used to model systems in which multiple factors affect the change of a quantity.
Differential equations are applied in many scientific and engineering fields, including physics, biology, chemistry, and economics. Some specific applications include predicting weather patterns, designing control systems for vehicles, and modeling chemical reactions in a lab setting.