Problem 03 | Laplace Transform of Derivatives

Problem 03
Find the Laplace transform of   f(t)=e5t   using the transform of derivatives.
 

Solution 03
f(t)=e5t       ..........       f(0)=1

f(t)=5e5t
 

L{f(t)}=sL{f(t)}f(0)

L(5e5t)=sL(e5t)1

1=sL(e5t)L(5e5t)

sL(e5t)5L(e5t)=1

Problem 02 | Laplace Transform of Derivatives

Problem 02
Find the Laplace transform of   f(t)=sin2t   using the transform of derivatives.
 

Solution 02
f(t)=sin2t       ..........       f(0)=0

f(t)=2sintcost=sin2t
 

L{f(t)}=sL{f(t)}f(0)

L(sin2t)=sL(sin2t)0

2s2+22=sL(sin2t)

sL(sin2t)=2s2+4

Problem 01 | Laplace Transform of Derivatives

Problem 01
Find the Laplace transform of   f(t)=t3   using the transform of derivatives.
 

Solution 01
f(t)=t3       ..........       f(0)=0

f(t)=3t2       ..........       f(0)=0

f(t)=6t       ..........       f(0)=0

f(t)=6
 

L{f(t)}=s3L{f(t)}s2f(0)sf(0)f(0)

Laplace Transform of Derivatives

For first-order derivative:
L{f(t)}=sL{f(t)}f(0)
 

For second-order derivative:
L{f(t)}=s2L{f(t)}sf(0)f(0)
 

For third-order derivative:
L{f(t)}=s3L{f(t)}s2f(0)sf(0)f(0)
 

For nth order derivative:

L{fn(t)}=snL{f(t)}sn1f(0)sn2f(0)fn1(0)

 

Problem 11 | Integrating Factors Found by Inspection

Problem 11
y(x2+y21)dx+x(x2+y2+1)dy=0
 

Solution 11
y(x2+y21)dx+x(x2+y2+1)dy=0

y(x2+y2)dxydx+x(x2+y2)dy+xdy=0

[y(x2+y2)dx+x(x2+y2)dy](ydxxdy)=0

(x2+y2)(ydx+xdy)(ydxxdy)=0

(ydx+xdy)ydxxdyx2+y2=0

d(xy)d[arctan(y/x)]=0

d(xy)d[arctan(y/x)]=0

Problem 06 - 07 | Integrating Factors Found by Inspection

Problem 06
y(y2+1)dx+x(y21)dy=0

Solution 06
y(y2+1)dx+x(y21)dy=0

y3dx+ydx+xy2dyxdy=0

(xy2dy+y3dx)+(ydxxdy)=0

y2(xdy+ydx)+(ydxxdy)=0

(xdy+ydx)+(ydxxdyy2)=0

d(xy)+d(xy)=0

d(xy)+d(xy)=0

xy+xy=c

xy2+x=cy

Problem 05 | Integrating Factors Found by Inspection

Problem 05
y(x4y2)dx+x(x4+y2)dy=0
 

Problem 05
y(x4y2)dx+x(x4+y2)dy=0

x4ydxy3dx+x5dy+xy2dy=0

(x4ydx+x5dy)+(xy2dyy3dx)=0

x4(ydx+xdy)+y2(xdyydx)=0

(ydx+xdy)+y2(xdyydx)x4=0

(ydx+xdy)+y2x2(xdyydxx2)=0

Compound Interest

In compound interest, the interest earned by the principal at the end of each interest period (compounding period) is added to the principal. The sum (principal + interest) will earn another interest in the next compounding period.
 

Consider $1000 invested in an account of 10% per year for 3 years. The figures below shows the contrast between simple interest and compound interest.
 

Derivation of Formula for the Future Amount of Ordinary Annuity

The sum of ordinary annuity is given by
 

F=A[(1+i)n1]i

 

To learn more about annuity, see this page: ordinary annuity, deferred annuity, annuity due, and perpetuity.
 

Derivation

Figure for Derivation of Sum of Ordinary Annuity

 

F= Sum

F=A+F1+F2+F3++Fn1+Fn

F=A+A(1+i)+A(1+i)2+A(1+i)3++A(1+i)n1+A(1+i)n