Question

dy 2. Find the general solution of -y+e"y dv -6xy 3. Find the general solution of t dr 4y+9x2 dy dx Find the general solution of бх2e" + 4y. 4. 5. Find the general solution of dr (y +2) dy 5x +4y Find the general solution of dx 8y3 d By'-4x

Answer 1

dy/dx = y + e^2x y^3 doublerightarrow dy/dx -y = e^2x y^3 Dividing by y^3 1/y^3 dydx - 1/y^2 = e^2x rightarrow (1) put -1/y^2 = v doublerightarrow 2/y^3 dy = dv 1/y^3 dy/dx = 1/2 dv/dx eqn (1) becomes 1/2 dv/dx + v = e^2x dv/dx + 2v = 2e^2x which is linear DE where P = 2 & Q = 2e^2x solution i.e of the form v e^integral p dx = integral e^integral p dx middot Q dx + C -1/y^2 e^integral 2 dx = integral e^integral 2 dx middot 2e^2x dx + C -1/y^2 e^2x = 2 integral e^2x e^2x dx + C \r\n -1/y^2 e^2x = 2 integral e^4x dx + C -1/y^2 e^2x = 2 e^4x/4 + C -1/y^2 = 1/2 e^2x + C e^-2x 1/2 e^2x + 1/y^2 + C e^-2x = 0 dy/dx = -6xy/4y + 9x^2 doublerightarrow (4y + 9x^2) dy = -6xy dx \r\n < null >