Question

У"+16у=0. Знайти загальний розв'язок ЛНДР

Answer 1

Y"=-16*у
у'=-8*у^2+constanta
y=(-8*y^3)/3+constanta

Answer 2

Solving the Linear Diophantine Equation (LDE)

To solve the linear Diophantine equation (LDE) 4x + 16y = 0 and find the general solution, we can use the extended Euclidean algorithm. The general form of a linear Diophantine equation is ax + by = c, where a, b, and c are integers.

Extended Euclidean Algorithm

The extended Euclidean algorithm is used to find the greatest common divisor (GCD) of two numbers and express it as a linear combination of the numbers. In this case, we want to find integers x and y such that 4x + 16y = 0.

Finding the General Solution

Using the extended Euclidean algorithm, we can find the general solution for the given LDE. The general solution for the LDE 4x + 16y = 0 is given by:

x = -4t y = t

where t is an integer.

Conclusion

In conclusion, the general solution to the linear Diophantine equation 4x + 16y = 0 is x = -4t and y = t, where t is an integer. This solution satisfies the given equation and represents all possible integer solutions for x and y that satisfy the equation.

I hope this helps! If you have any further questions or need additional assistance, feel free to ask.