Question

dy/dx = sin(x+y) solve for differential equation

Answer 1

take x +y =t then it will solve easily bro

Answer 2

Answer:

The required equation is  $\tan (x+y)-\sec (x+y)=x+C$

Step-by-step explanation:

Given : Expression $\frac{dy}{dx}=\sin(x+y)$

To find : Solve for differential equation?

Solution :

$\frac{dy}{dx}=\sin(x+y)$………..(1)

Let, x+y=u

Differentiate with respect to x,

$1+\frac{dy}{dx}=\frac{du}{dx}$

$\frac{dy}{dx}=\frac{du}{dx}-1$ .........(2)

From equation (1) and (2),

$\frac{du}{dx}-1 =sin u$

$\frac{du}{dx}=sin u+1$

$(\frac{1}{1+\sin u})du=dx$

$\frac{1-\sin u}{(1+\sin u)(1-\sin u)}du=dx$

$\frac{1-\sin u}{\cos^2u}du=dx$

$(\sec^2 u - \sec u.\tan u)du=dx$

Integrate both side,

$\tan u-\sec u=x+C$  

$\tan (x+y)-\sec (x+y)=x+C$

Therefore, The required equation is  $\tan (x+y)-\sec (x+y)=x+C$