promath The integral of e^(-x^2) from 0 to infinity is equal to √π/2, that is, ∫exp(-x^2) dx = √π/2. Here we will learn how to integrate e-x^2 (e to the power -x2) from 0 to ∞. Integration of exp(-x^2) from 0 to infinity Answer: The integral of $e^{-x^2}$ from 0 to infinity is equal to √π/2. … Read more
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
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
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
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
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
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
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
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
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