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Laplace Transform of sint/t | Laplace of sin(at)/t

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The Laplace transform of sint/t is equal to tan-1(1/s), that is, L{sint/t} = tan-1(1/s). The formulas for the Laplace of sint/t and the Laplace of sin(at)/t are given as follows: $\boxed{\mathcal{L}\left\{\dfrac{\sin t}{t} \right\}=\tan^{-1}\left(\dfrac{1}{s} \right)}$ and $\boxed{\mathcal{L}\left\{\dfrac{\sin at}{t} \right\}=\tan^{-1}\left(\dfrac{a}{s} \right)}$ Lets learn how to find Laplace of sin(t)/t. Laplace of sint/t Question: What is the Laplace … Read more

Solve dy/dx+y/x=x^2y^6 [Bernoulli Differential Equation]

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In this page, we solve dy/dx+y/x=x^2y^6. This is a Bernoulli’s differential equation. The general solution of dy/dx+y/x=x^2y^6 is 5x3y5 + Cx5y5 = 2 where C denotes an arbitrary constant of integrals. Let us now solve the Bernoulli’s differential equation dy/dx+y/x=x2y6. Solution of dy/dx+y/x=x^2y^6 Question: Find the solution of $\dfrac{dy}{dx}+\dfrac{y}{x}=x^2y^6$. Answer: $\dfrac{dy}{dx}+\dfrac{y}{x}=x^2y^6$ …(I) The given differential … Read more

How to Solve (D^2+4)y=cos^2x

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The general solution of (D2+4)y=cos2x is given by y=c1cos2x + c2sin2x + (1+xsin2x)/8. Lets learn how to solve (D^2+4)y=cos^2x. As D2 = d2/dx2, the differential equation (D2+4)y=cos2x can also be written as follows: d2y/dx2 + 4y = cos2x. Solve (D2+4)y=cos^2x Question: Solve the equation $\dfrac{d^2y}{dx^2} + 4y = \cos^2x$. Solution: Letting D2 = d2/dx2, we … Read more

How to Solve (D^2+4)y=sin^2x

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Here we learn how to solve (D^2+4)y=sin^2x. The general solution of (D2+4)y=sin2x is given by y=c1cos2x + c2sin2x + (1-xsin2x)/8. The differential equation (D2+4)y=sin2x can also be written as follows: d2y/dx2 + 4y = sin2x. Now let us solve this equation. Solve (D2+4)y=sin^2x Question: Solve the equation $\dfrac{d^2y}{dx^2} + 4y = \sin^2x$. Solution: One can … Read more

Solve the Equation d^2y/dx^2 + 4y = cos2x | (D^2+4)y=cos2x

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Here we solve the equation d^2y/dx^2+4y=cos2x, that is, Solve (D^2+4)y=cos2x. The general solution of d2y/dx2 + 4y = cos2x is equal to y=c1cos2x + c2sin2x + (xsin2x)/4. Solve (D2+4)y=cos2x Question: Solve the equation $\dfrac{d^2y}{dx^2}$ + 4y = cos2x. Solution: Letting D2 = d2/dx2, we can write the given differential equation as follows: (D2+4)y = cos2x … Read more

Solve the Equation d^2y/dx^2 + 4y = sin2x | Solve (D^2+4)y=sin2x

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Here we solve (D^2+4)y=sin2x. The general solution of d2y/dx2 + 4y = sin2x is given by y=c1cos2x + c2sin2x – (xcos2x)/4. In this page, let us learn to solve the equation d^2y/dx^2+4y=sin2x, i.e, (D^2+4)y=sin2x. Solve (D2+4)y=sin2x Question: Solve the equation $\dfrac{d^2y}{dx^2}$ + 4y = sin2x. Solution: The given differential equation can be written in the … Read more

Derivative of ln(ln(x)) | Differentiate ln(ln(lnx))

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The derivative of ln(ln(x)) is equal to 1/{x lnx ln(lnx)}. This is obtained by using the chain rule of differentiation. In this post, lets learn how to differentiate ln(ln(lnx)). Differentiation of ln(ln(lnx)) Step 1: Recall, the chain rule of differentiation: Suppose $f$ is a function of $z$ and $z$ is a function of $x$, that … Read more

Difference Between Convergent and Divergent Sequence

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A sequence is called convergent if its limit is finite, and divergent if its limit is infinite (either +∞ or -∞). The difference between these two type of sequences will be discussed in this page. Convergent and Divergent Sequence Difference The differences of convergent and divergent sequences are listed in the table below. Convergent Squence … Read more

Divergent Sequence: Definition, Examples

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A divergent sequence is a sequence having limit either infinite (that is, does not converge to a specific finite limit). For example, the sequence {n} has infinite limit, hence a divergent sequence. Here we learn the definition and examples of divergent sequence. Divergent Sequence Definition A sequence {xn} is said be divergent if its limit … Read more

Convergent Sequence: Definition and Examples

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A convergent sequence is a sequence having finite limit. For example, {1/n} is a convergent sequence with limit 0. Here, we study the definition and examples of convergent sequence. Convergent Sequence Definition A sequence {xn} is said be convergent if it has a finite limit L. That is, for every ε>0, there is a natural … Read more