dy/dx=sin(x+y) Solve the Differential Equation [Solution]

The general solution of the differential equation dy/dx=sin(x+y) is given by tan(x+y) – sec(x+y) = x +C where C is an arbitrary constant. Here we will learn how to find the solution of dy/dx=sin(x+y).

Let us now solve the differential equation dy/dx=sin(x+y) by the substitution method.

General Solution of dy/dx=sin(x+y)

Question: Find the general solution of dy/dx=sin(x+y).

Answer:

Step 1: Use the substitution z=x+y.

Differentiating w.r.t x, we get

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

⇒ $\dfrac{dy}{dx}=\dfrac{dz}{dx}-1$

Step 2: Use the above data in the given differential equation which is dy/dx=sin(x+y). Thus, we deduce that

$\dfrac{dz}{dx}$ -1 = sin z

⇒ dz/dx = 1+sinz

Step 3: Separate the variables x and z and rewrite the above equation. Therefore,

$\dfrac{dz}{1+\sin z}=dx$

Step 4: Integrating both sides of the above equation,

∫ $\dfrac{dz}{1+\sin z}$ = ∫dx +C

Integrating both sides, we get that

⇒ ∫ $\dfrac{1-\sin z}{(1+\sin z)(1-\sin z)} dz$ = ∫dx +C

⇒ ∫ $\dfrac{1-\sin z}{1-\sin^2 z} dz$ = ∫dx +C

⇒ ∫ $\dfrac{1-\sin z}{\cos^2 z} dz$ = ∫dx +C

⇒ ∫ sec²z dz – ∫secz tanz dz = ∫dx +C

⇒ tan z – sec z = x +C

⇒ tan(x+y) – sec(x+y) = x +C

So the general solution of dy/dx=sin(x+y) is equal to tan(x+y) – sec(x+y) = x +C where C is an integral constant.

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