Julio said:Solve the Cauchy problem:
\begin{align}
\dfrac{dx}{dr} &= y\\
x(0,s) &= s
\end{align}
Help please, I don't remember how solve this :(. Separate variable isn't the method?
The Cauchy problem is a type of initial value problem in mathematics that involves finding a solution to a differential equation with given initial conditions. It is named after the French mathematician Augustin-Louis Cauchy.
The separate variable method is a technique used to solve differential equations by separating the dependent and independent variables on opposite sides of the equation. This allows for the integration of each side separately, making it easier to find a solution.
To solve a Cauchy problem using the separate variable method, you must first separate the dependent and independent variables on opposite sides of the equation. Then, integrate each side separately and solve for the unknown variable. Finally, use the initial conditions given in the problem to determine the constant of integration and obtain the final solution.
The key steps in solving a Cauchy problem using the separate variable method are: 1) separating the dependent and independent variables, 2) integrating each side separately, 3) solving for the unknown variable, and 4) using the initial conditions to determine the constant of integration and obtain the final solution.
Some common mistakes to avoid when using the separate variable method to solve a Cauchy problem include: 1) forgetting to separate the variables on opposite sides of the equation, 2) making errors while integrating each side separately, 3) forgetting to include the constant of integration, and 4) not using the initial conditions to determine the final solution.