Solving y'' = -y with y(0) = 0, y(π) = 1

Differential Equations · Axiom Academy

Apply the shooting method to a well-posed boundary value problem Solve the boundary value problem below by the shooting method — guess the missing initial slope, integrate the resulting initial value problem, and adjust the guess until the far boundary condition is met. Nice work — you applied the shooting method to a genuine boundary value problem and watched it converge. Here's what carries forward: BVPs vs IVPs: a boundary value problem fixes y at two different points instead of fixing y and y' at the same point, so you can't just integrate forward — you first have to find the right starting slope. The Shooting Method: convert the BVP into a sequence of IVPs by guessing the missing initial slope, integrating (RK4 here), and measuring the residual at the far boundary. Bracket, then refine: one guess that undershoots and one that overshoots bracket the root; a secant update on the two residuals lands on the correct slope in very few iterations — here, exactly s^ * =1 . Well-posedness matters: this only works because the target point is where , so the boundary condition genuinely pins down B . (Placing the second condition at instead — where for every choice of B — would make the problem unsolvable, and no amount of shooting could ever converge.) The shooting method is a core numerical technique for boundary value problems — you'll use it whenever an exact solution is hard to reach analytically, in physics, engineering, and applied mathematics alike.

This is the written version of the interactive lesson above. See the full Differential Equations course.