Find General Solution of dy/dx=1+x+y+xy

The solution of the differential equation dy/dx=1+x+y+xy is equal to y= 1 – $C e^{x +\frac{x^2}{2}}$ where C is an arbitrary constant. In this post, we will learn how to solve dy/dx=1+x+y+xy.

Solution of dy/dx=1+x+y+xy

Question: Solve the differential equation $\dfrac{dy}{dx}$ = 1+x+y+xy.

Solution:

We will solve solve $\dfrac{dy}{dx}$ =1+x+y+xy by variable separable method. The given differential equation can be written as follows:

$\dfrac{dy}{dx}$ = (1+x) + y(1+x)

⇒ $\dfrac{dy}{dx}$ = (1+x) (1+y)

⇒ $\dfrac{dy}{1+y} = (1+x) dx$

Integrating we get that

$\int \dfrac{dy}{1+y} = \int (1+x) dx +k$

⇒ $\ln |1+y| = x +\dfrac{x^2}{2} +k$ by the power rule of integration: ∫xⁿ dx = xⁿ⁺¹/(n+1).

⇒ $1+y= C e^{x +\frac{x^2}{2}}$ where C=e^k.

⇒ $y= 1-C e^{x +\frac{x^2}{2}}$

So the general solution of the differential equation dy/dx= 1+x+y+xy is given by $y= 1-C e^{x +\frac{x^2}{2}}$.

Remark: For the above we see that the exact solution of the differential equation dy/dx = (1+x)(1+y) is equal to y= 1 – $C e^{x +\frac{x^2}{2}}$.

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