Which of the following is a homogenous first order differential equation dy/dx=y

Question

Which of the following is a homogenous first order differential equation dy/dx=y^2-xy/x^2 dy/dx=xy^2-y^3/x^3+y^2 dy/dx = x^2y/x^3-3xy^3 dy/dx = y^2-x^2 Which of the differential equations is separable All three are separable Equation (i) only Equation (ii) only Equation (i) and (iii) only Use the separation of the variable to find the general solution of the differential equation 1/2 tan^-1 y^2 = sinx + C In(1+y^4) = -4sinx + C -1/2y^2 + y^2/2 = sinx + C In(1+y^4) = 4sinx + C What is the integrating factor, I(x) for the following linear first order differential equation I(x) = 1/x^2 I(x) = e^-2x I(x) = -x^2 I(x) = -2x y_1 = e^-2x and y_2 and e^x are solutions for: y" - y' - 2y = 0 y" + y' - 2y = 0 y" - y' - 2y' - 3y = 0 y" + y' +2y = 0

Explanation / Answer

1. a) y'= y^2 -xy /x^2

= (y/x)^2 - (y/x)

So, we can see the differential equation is of the form (y/x) , so it is first order differntail equation.

2. Equation i and iii are separable :

i) y'= -xy

y''/y = -x ---> dy/y = -xdx

iii) y'= xy +x

dy/(1+y) = xdx

5. y1 = e^-2x ; y2 = e^x

B : y"+ y'-2y=0

roots are (m +2)(m-1) =0

m = -2 ; m=1

which is the given general solution in question

So, Option B

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