Question: Consider the Sturm-Liouville problem dy λ d[x] + ² y = x y' (1) = 0 y'(e²¹) = = 0 Find the characteristic functions if there exists. Show that they are orthogonal with respect to weight function.

Q: Find y' and y". y = In(x + √5 + x² ² = In ( x +

A: Given:  y=lnx+5+x2   To find the value of y', y''.  The formula of ln x'=1x. The formula of…

Q: Find the implicit deriviate (dy/dx) of (2x+3y)1/3=x2

A:

Q: 2 y = √√√x² + 1

A:

Q: 3. Find the equations of the tangent and normal lines to the curve f(x) = /2* +cos(x*) at(0,/2).…

A:

Q: 6. Find * : y- + 5)ao TK, (b) y = Sin 5x I+cos 5x dx

A: Solving the problem by finding the derivative.

Q: How can y = ex in the vicinity of x = 0 (with appropriate labels) be sketched?

A: To sketch the graph of y=ex in the vicinity of x=0, follow these steps: 1. Understand the Basic…

Q: o(e) = La t -Va V1 + t²

A:

Q: b.9f Fex)=x Sin lgix), where gL1)= , and g' (1)= -1, then f'(1)=2

A:

Q: To identify how much heat is required to bring a kettle of water to its boiling point, you are…

A: Here,

Q: 8) a) Find dy/dx if: e3x = x+y b) Differenctiate y = (sin r)"

A: We know that general derivative of d(xn)/dx is nxn-1. And derivative of d(sinx)/dx = cosx  d(lnx)/dx…