2.6.6—Initial conditions

Solving an ODE gives a family containing arbitrary constants. Initial conditions such as y(0)=a and y′(0)=b provide simultaneous equations that determine those constants.

Keep the general solution and its derivative visible, substitute the conditions at the same point, and solve the resulting linear system. For a second-order equation, two independent conditions are normally needed.

If y=C₁e^x+C₂e^{2x}, y(0)=3 gives C₁+C₂=3; y′(0)=4 gives C₁+2C₂=4, so C₁=2 and C₂=1.

One condition cannot determine two arbitrary constants, and applying a condition before differentiating can lose necessary information.