Question

Найти общее решение d^2y/dx^2-2*dy/dx+5y=0

Answer 1

Единицу вправо, два раза интегрируем
dy/dx = x + C1; С1 = 0
y = x^2/2 + С2; С2 = 2
Итого y = x^2/2 + 2

Answer 2

Finding the General Solution of d^2y/dx^2 - 2(dy/dx) + 5y = 0

To find the general solution of the differential equation d^2y/dx^2 - 2(dy/dx) + 5y = 0, we can use the method of characteristic equations.

Let's denote dy/dx as p and d^2y/dx^2 as q. The given differential equation can be rewritten as:

q - 2p + 5y = 0

Now, we can write the characteristic equation as:

r^2 - 2r + 5 = 0

Solving this quadratic equation, we find the roots:

r1 = 1 + 2i r2 = 1 - 2i

Since the roots are complex, we can write the general solution as:

y(x) = e^x(C1*cos(2x) + C2*sin(2x)) where C1 and C2 are arbitrary constants.

Please note that the source provides the general solution for a similar differential equation, which can be applied to the given equation.

I hope this helps! Let me know if you have any further questions.