Question

Find and sketch the curve y(x) which passes through the points (0,1),(1,6​), and where the functional J[y(x)]=∫01​y²(y′)²+y²dx is extremized.

Answer 1

To find and sketch the curve y(x) that passes through the given points and extremizes the functional J[y(x)]=∫[0 to 1] y²(y')² + y² dx, we can use the Euler-Lagrange equation. This equation can be solved to find the curve y(x).

Step-by-step explanation:

To find and sketch the curve y(x) that passes through the points (0,1) and (1,6) and extremizes the functional J[y(x)]=∫[0 to 1] y²(y')² + y² dx, we need to use the Euler-Lagrange equation. Let's start by finding the Euler-Lagrange equation:

L = y²(y')² + y²

dL/dy - d/dx(dL/dy') = 0:

2yy'² + 2y - 2yy'' - 2(y')³ = 0

Simplifying further, we get:

yy'² - y - yy'' - (y')³ = 0

We can now solve this second-order nonlinear differential equation to obtain the curve y(x).