Question

Найти общее решение дифференциального уравнения у(штрих) + у/х=1

Answer 1

Y'=dy/dx
dy/dx+(1/x)*y=1
u(x)=e^integral(1/x)*dx=x
x*(dy/dx)+x*(1/x)*y=x*1
x*(dy/dx)+y=x
x*(dy/dx)=x
y*x=integral x*dx
y*x=(x^2/2)+C, C=const
y=((x^2/2)+C)/x
y=(x^2+2*C)/2x
y=x/2+C/x, C=const

Answer 2

General Solution of the Differential Equation

The given differential equation is:

u'(x) + u(x)/x = 1

To find the general solution of this differential equation, we can use an integrating factor. The integrating factor for this equation is x. Multiplying both sides of the equation by x gives:

xu'(x) + u(x) = x

Now, we can rewrite this equation as:

(xu(x))' = x

Integrating both sides with respect to x gives:

∫(xu(x))' dx = ∫x dx

Using the fundamental theorem of calculus, we can simplify the integrals:

xu(x) = (1/2)x^2 + C

Dividing both sides by x gives the general solution:

u(x) = (1/2)x + C/x

where C is the constant of integration.

Therefore, the general solution of the given differential equation is u(x) = (1/2)x + C/x.

Please note that this solution is obtained by integrating the given differential equation and is valid for all values of x except x = 0.

Let me know if you need any further assistance!