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Solve the differential equation $v^{\prime}-2=\cos\left(v\right)$

Step-by-step Solution

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Solution

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Final answer to the problem

$2\cdot \left(\frac{1}{\sqrt{3}}\right)\arctan\left(\frac{\tan\left(\frac{v}{2}\right)}{\sqrt{3}}\right)=x+C_0$
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Step-by-step Solution

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  • 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
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1

Rewrite the differential equation using Leibniz notation

Learn how to solve integrals of rational functions of sine and cosine problems step by step online.

$\frac{dv}{dx}-2=\cos\left(v\right)$

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Learn how to solve integrals of rational functions of sine and cosine problems step by step online. Solve the differential equation v^'-2=cos(v). Rewrite the differential equation using Leibniz notation. Grouping the terms of the differential equation. Group the terms of the differential equation. Move the terms of the v variable to the left side, and the terms of the x variable to the right side of the equality. Integrate both sides of the differential equation, the left side with respect to y, and the right side with respect to x.

Final answer to the problem

$2\cdot \left(\frac{1}{\sqrt{3}}\right)\arctan\left(\frac{\tan\left(\frac{v}{2}\right)}{\sqrt{3}}\right)=x+C_0$

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Function Plot

Plotting: $2\cdot \left(\frac{1}{\sqrt{3}}\right)\arctan\left(\frac{\tan\left(\frac{v}{2}\right)}{\sqrt{3}}\right)=x+C_0$

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n
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v
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x
y
z
.
(◻)
+
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×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
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

How to improve your answer:

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Main Topic: Integrals of Rational Functions of Sine and Cosine

The integrals of rational functions of sine and cosine are integrals where the integrand is a rational function composed of sines and cosines, usually in the denominator, which is not easily reducible. However, these integrals are easy to solve by applying the recommended substitution: $z=\tan\left(\frac{x}{2}\right)$

Used Formulas

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