(x+y+1)dy/dx=1 Find the General Solution | (x+y+1)dy/dx=1 Solution

The general solution of the differential equation (x+y+1)dy/dx=1 is equal to x= -(y+2)+Ce^y where C denotes an arbitrary constant. Here we will learn how to solve the differential equation (x+y+1)dy/dx=1.

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

Question: What is the solution of (x+y+1)$\dfrac{dy}{dx}$=1?

Answer:

Let us put x+y+1 = v.

Differentiating w.r.t x, we get that

$1+\dfrac{dy}{dx}=\dfrac{dv}{dx}$

⇒ $\dfrac{dy}{dx}=\dfrac{dv}{dx}-1$

So the given equation becomes

$v(\dfrac{dv}{dx}-1)=1$

⇒ $\dfrac{dv}{dx}=\dfrac{1}{v}+1$

⇒ $\dfrac{dv}{dx}=\dfrac{1+v}{v}$

⇒ $\dfrac{v}{1+v}dv=dx$

Integrating, $\int \dfrac{v}{1+v}dv=\int dx+K$

$\int \dfrac{1+v-1}{1+v}dv=x+K$

⇒ $\int dv -\int \dfrac{dv}{1+v}=x+K$

⇒ $v -\log|1+v|=x+K$

⇒ $1+x+y -\log|1+x+y+1|=x+K$

⇒ $1+y -\log|x+y+2|=K$

⇒ $\log|x+y+2|=y+k$ where k= -K+1

⇒ $x+y+2=Ce^y$ where C= e^k

⇒ $x=-(y+2)+Ce^y$

So the solution of (x+y+1)dy/dx=1 is given by x= -(y+2)+Ce^y where C is an integral constant.

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