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# Solve the differential equation $\frac{dy}{dx}=\tan\left(x+y\right)$

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##  Final answer to the problem

$-\frac{1}{2}\ln\left(1+\tan\left(\frac{x+y}{2}\right)^{2}\right)+\arctan\left(\tan\left(\frac{x+y}{2}\right)\right)+\frac{1}{2}\ln\left(1-\tan\left(\frac{x+y}{2}\right)^{2}+2\tan\left(\frac{x+y}{2}\right)\right)=x+C_0$
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##  Step-by-step Solution 

How should I solve this problem?

• Choose an option
• Exact Differential Equation
• Linear Differential Equation
• Separable Differential Equation
• Homogeneous Differential Equation
• Integrate by partial fractions
• Product of Binomials with Common Term
• FOIL Method
• Integrate by substitution
• Integrate by parts
Can't find a method? Tell us so we can add it.
1

When we identify that a differential equation has an expression of the form $Ax+By+C$, we can apply a linear substitution in order to simplify it to a separable equation. We can identify that $x+y$ has the form $Ax+By+C$. Let's define a new variable $u$ and set it equal to the expression

$u=x+y$

Learn how to solve equations problems step by step online.

$u=x+y$

Learn how to solve equations problems step by step online. Solve the differential equation dy/dx=tan(x+y). When we identify that a differential equation has an expression of the form Ax+By+C, we can apply a linear substitution in order to simplify it to a separable equation. We can identify that x+y has the form Ax+By+C. Let's define a new variable u and set it equal to the expression. Isolate the dependent variable y. Differentiate both sides of the equation with respect to the independent variable x. Now, substitute x+y and \frac{dy}{dx} on the original differential equation. We will see that it results in a separable equation that we can easily solve.

##  Final answer to the problem

$-\frac{1}{2}\ln\left(1+\tan\left(\frac{x+y}{2}\right)^{2}\right)+\arctan\left(\tan\left(\frac{x+y}{2}\right)\right)+\frac{1}{2}\ln\left(1-\tan\left(\frac{x+y}{2}\right)^{2}+2\tan\left(\frac{x+y}{2}\right)\right)=x+C_0$

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

SnapXam A2

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0
a
b
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f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

###  Main Topic: Equations

In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. In this situation, variables are also known as unknowns and the values which satisfy the equality are known as solutions.