3.8.2—Differential equations

Integrating an ODE introduces arbitrary constants. The order of the equation indicates how many independent conditions are normally needed to select one solution.

Keep the general solution visible, use conditions at the specified point, and check the resulting function in both the differential equation and the conditions.

Solving y′=2x gives y=x²+C; y(0)=3 selects y=x²+3.

A general solution is not one trajectory, and a condition on y′ cannot replace a missing condition on y when only one constant exists.