promath 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
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
An unbounded sequence is a sequence which is not bounded. For example the sequence {n} of natural numbers is unbounded. In this article, we study unbounded sequences with its definition and examples. Unbounded Sequence Definition A sequence {an} is said to be unbounded if there is no real number M such that |an| ≤ M. … Read more
No, a continuous function is not always differentiable. For example, f(x)=|x| is continuous but not differentiable at x=0. In this post, we will study the following: is a continuous function always differentiable? A continuous function may not be differentiable always. Although, a differentiable function is always continuous. Lets prove this fact here. Recall the definitions … Read more
The formula of 1-cos2x is given by 1-cos2x=2sin2x. In this post, we establish 1-cos2x formula and identity with some solved examples. Formula of 1-cos2x Answer: The 1-cos2x formula is 1-cos2x=2sin2x. Proof: Using the formula cos2θ= cos2θ -sin2θ, the given expression can be written as follows. 1-cos2x = 1- (cos2x -sin2x) = (1- cos2x) -sin2x = … Read more
In this article, we prove that a convergent sequence is bounded. A sequence is said to be convergent if it has finite limit. Question: Prove that A Convergent Sequence is Bounded. Proof Let us assume that {xn} is a convergent sequence. So it has finite limit, say L. That is, limn→∞ xn = L. So … Read more