To solve the second order differential equation given by yy – 12y” = 0, we start by rewriting it in a standard form.
The equation can be rearranged to:
12y” = yy
By dividing both sides by 12, we find:
y” = rac{yy}{12}
This is a nonlinear second order differential equation. One standard method to approach such equations is to assume a solution of the form y = y(x). Let’s try to find solutions by substitution and analysis.
1. **Finding the characteristic equation:** We consider the case where the solution could be polynomial or exponential. However, due to the nonlinearity, we will check for simple solutions first.
2. **Check for simple solutions:** A straightforward approach can be to check for constant solutions. Let’s assume y = C, where C is a constant:
y” = 0
Then the equation becomes:
0 = C^2
So the only constant solution is C = 0.
3. **Finding non-constant solutions:** Next, let’s look for solutions in a power series or consider functions like polynomials or other simple applications of differential equations.
Through methods of undetermined coefficients or power series or possibly applying the substitution methods, we notice that the general character of the solutions can be complex and involve nonlinear dynamics.
4. **Using a proper substitution:** If we assume y = vx, where v is a function of x, we can transform the equation. However, this will require substituting for derivatives and forming a new differential equation.
5. **Final remarks:** Due to the complexity of the second-order nonlinear nature of the equation, sometimes numerical methods or specific functional forms can give insights. The general solution may not have a simple closed form and can depend deeply on initial conditions and boundary properties of the system.
Thus, analyzing the behaviors of possible forms through graphing or further transforming methods would help illustrate the solution trajectories, leading to various possible non-linear behaviors as x varies.