An ODE (ordinary differential equation) is a mathematical equation that describes the relationship between a function and its derivatives. In this case, the ODE is dy/dx = (x+y)^2.
To solve a first-order ODE, we can use a variety of methods such as separation of variables, integrating factors, or using the method of undetermined coefficients. In this case, we can use the method of separation of variables to solve dy/dx = (x+y)^2.
The initial condition, y(0)=1, tells us the value of the function y at the point x=0. It is an important piece of information that helps us find the specific solution to the ODE.
To find the general solution to this ODE, we first rearrange the equation to separate the variables x and y. Then, we integrate both sides and add a constant of integration. This will give us a general solution that includes all possible solutions to the ODE.
The specific solution represents the particular solution to the ODE that satisfies the initial condition. In this case, the specific solution is y = tan(x+C) where C is the constant of integration.