Find the general solution for differential equations. (x + y + 1) dy/dx = 1

Question

Find the general solution for differential equations.

 (x + y + 1) \(\frac {dy}{dx}\) = 1

(x + y + 1) dy/dx = 1

Answer 1

Given Differential Equation :

(x + y + 1) \(\frac {dy}{dx}\) = 1

Formula :

i) \(\int\) 1dx = x

ii) \(\int\) u. v dx = u. \(\int\) v. dx - \(\int\) \((\frac{du}{dx}.\int v\, dx)\) dx

iii) \(\int\) e^kx dx = \(\frac {e^{kx}}{k}\)

iv) \(\frac{d}{dx}\) (xⁿ) = nx^n-1

v) General solution :

For the differential equation in the form of

\(\frac{dx}{dy} \,+ Px\, =Q\)

General solution is given by,

x. (I.F.) = \(\int\) Q. (I.F.) dy + c

Where, integrating factor,

I.F. = \(e^{\int p\, dx}\)

Given differential equation is

Equation (1) is of the form

\(\frac{dx}{dy} \,+ Px\, =Q\)

where, P= - 1 and Q = y + 1

Therefore, integrating factor is

General solution is

Let, u=y+1 and v= e^-y

Substituting I in eq(2),

Dividing above equation by e^-y

Therefore, general solution is

x ce^y - (y + 2)