What is the general solution for solvefor x,(D^2-3D+2)y=sec^2(e^{-x}) ?

The general solution for solvefor x,(D^2-3D+2)y=sec^2(e^{-x}) is

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solvefor x,(D^2-3D+2)y=sec^2(e^{-x})

Solution

solve for

x, (D^{2} - 3 D + 2) y = sec^{2} (e^{- x})

Solution

x = - ln (arcsec (y (D^{2} - 3 D + 2)) + 2 πn), x = - ln (- arcsec (y (D^{2} - 3 D + 2)) + 2 πn), x = - ln (arcsec (- y (D^{2} - 3 D + 2)) + 2 πn), x = - ln (- arcsec (- y (D^{2} - 3 D + 2)) + 2 πn)

Solution steps

(D^{2} - 3 D + 2) y = sec^{2} (e^{- x})

Switch sides

sec^{2} (e^{- x}) = (D^{2} - 3 D + 2) y

Solve by substitution

sec^{2} (e^{- x}) = (D^{2} - 3 D + 2) y

Let:

sec (e^{- x}) = u

u^{2} = (D^{2} - 3 D + 2) y

u^{2} = (D^{2} - 3 D + 2) y : u = (D^{2} - 3 D + 2) y, u = - (D^{2} - 3 D + 2) y

u^{2} = (D^{2} - 3 D + 2) y

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = (D^{2} - 3 D + 2) y, u = - (D^{2} - 3 D + 2) y

Substitute back

u = sec (e^{- x})

sec (e^{- x}) = (D^{2} - 3 D + 2) y, sec (e^{- x}) = - (D^{2} - 3 D + 2) y

sec (e^{- x}) = (D^{2} - 3 D + 2) y, sec (e^{- x}) = - (D^{2} - 3 D + 2) y

sec (e^{- x}) = (D^{2} - 3 D + 2) y : x = - ln (arcsec (y (D^{2} - 3 D + 2)) + 2 πn), x = - ln (- arcsec (y (D^{2} - 3 D + 2)) + 2 πn)

sec (e^{- x}) = (D^{2} - 3 D + 2) y

Apply trig inverse properties

sec (e^{- x}) = (D^{2} - 3 D + 2) y

General solutions for

sec (e^{- x}) = (D^{2} - 3 D + 2) y

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

e^{- x} = arcsec ((D^{2} - 3 D + 2) y) + 2 πn, e^{- x} = - arcsec ((D^{2} - 3 D + 2) y) + 2 πn

e^{- x} = arcsec ((D^{2} - 3 D + 2) y) + 2 πn, e^{- x} = - arcsec ((D^{2} - 3 D + 2) y) + 2 πn

Solve

e^{- x} = arcsec ((D^{2} - 3 D + 2) y) + 2 πn : x = - ln (arcsec (y (D^{2} - 3 D + 2)) + 2 πn)

e^{- x} = arcsec ((D^{2} - 3 D + 2) y) + 2 πn

Apply exponent rules

e^{- x} = arcsec ((D^{2} - 3 D + 2) y) + 2 πn

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{- x}) = ln (arcsec ((D^{2} - 3 D + 2) y) + 2 πn)

Apply log rule:

ln (e^{a}) = a

ln (e^{- x}) = - x

- x = ln (arcsec ((D^{2} - 3 D + 2) y) + 2 πn)

- x = ln (arcsec ((D^{2} - 3 D + 2) y) + 2 πn)

Solve

- x = ln (arcsec ((D^{2} - 3 D + 2) y) + 2 πn) : x = - ln (arcsec (y (D^{2} - 3 D + 2)) + 2 πn)

- x = ln (arcsec ((D^{2} - 3 D + 2) y) + 2 πn)

Divide both sides by

- 1

- x = ln (arcsec ((D^{2} - 3 D + 2) y) + 2 πn)

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{ln ( arcsec ( ( D ^{2} - 3 D + 2 ) y ) + 2 πn )}{- 1}

Simplify

x = - ln (arcsec (y (D^{2} - 3 D + 2)) + 2 πn)

x = - ln (arcsec (y (D^{2} - 3 D + 2)) + 2 πn)

x = - ln (arcsec (y (D^{2} - 3 D + 2)) + 2 πn)

Solve

e^{- x} = - arcsec ((D^{2} - 3 D + 2) y) + 2 πn : x = - ln (- arcsec (y (D^{2} - 3 D + 2)) + 2 πn)

e^{- x} = - arcsec ((D^{2} - 3 D + 2) y) + 2 πn

Apply exponent rules

e^{- x} = - arcsec ((D^{2} - 3 D + 2) y) + 2 πn

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{- x}) = ln (- arcsec ((D^{2} - 3 D + 2) y) + 2 πn)

Apply log rule:

ln (e^{a}) = a

ln (e^{- x}) = - x

- x = ln (- arcsec ((D^{2} - 3 D + 2) y) + 2 πn)

- x = ln (- arcsec ((D^{2} - 3 D + 2) y) + 2 πn)

Solve

- x = ln (- arcsec ((D^{2} - 3 D + 2) y) + 2 πn) : x = - ln (- arcsec (y (D^{2} - 3 D + 2)) + 2 πn)

- x = ln (- arcsec ((D^{2} - 3 D + 2) y) + 2 πn)

Divide both sides by

- 1

- x = ln (- arcsec ((D^{2} - 3 D + 2) y) + 2 πn)

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{ln ( - arcsec ( ( D ^{2} - 3 D + 2 ) y ) + 2 πn )}{- 1}

Simplify

x = - ln (- arcsec (y (D^{2} - 3 D + 2)) + 2 πn)

x = - ln (- arcsec (y (D^{2} - 3 D + 2)) + 2 πn)

x = - ln (- arcsec (y (D^{2} - 3 D + 2)) + 2 πn)

x = - ln (arcsec (y (D^{2} - 3 D + 2)) + 2 πn), x = - ln (- arcsec (y (D^{2} - 3 D + 2)) + 2 πn)

sec (e^{- x}) = - (D^{2} - 3 D + 2) y : x = - ln (arcsec (- y (D^{2} - 3 D + 2)) + 2 πn), x = - ln (- arcsec (- y (D^{2} - 3 D + 2)) + 2 πn)

sec (e^{- x}) = - (D^{2} - 3 D + 2) y

Apply trig inverse properties

sec (e^{- x}) = - (D^{2} - 3 D + 2) y

General solutions for

sec (e^{- x}) = - (D^{2} - 3 D + 2) y

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = - arcsec (a) + 2 πn

e^{- x} = arcsec (- (D^{2} - 3 D + 2) y) + 2 πn, e^{- x} = - arcsec (- (D^{2} - 3 D + 2) y) + 2 πn

e^{- x} = arcsec (- (D^{2} - 3 D + 2) y) + 2 πn, e^{- x} = - arcsec (- (D^{2} - 3 D + 2) y) + 2 πn

Solve

e^{- x} = arcsec (- (D^{2} - 3 D + 2) y) + 2 πn : x = - ln (arcsec (- y (D^{2} - 3 D + 2)) + 2 πn)

e^{- x} = arcsec (- (D^{2} - 3 D + 2) y) + 2 πn

Apply exponent rules

e^{- x} = arcsec (- (D^{2} - 3 D + 2) y) + 2 πn

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{- x}) = ln (arcsec (- (D^{2} - 3 D + 2) y) + 2 πn)

Apply log rule:

ln (e^{a}) = a

ln (e^{- x}) = - x

- x = ln (arcsec (- (D^{2} - 3 D + 2) y) + 2 πn)

- x = ln (arcsec (- (D^{2} - 3 D + 2) y) + 2 πn)

Solve

- x = ln (arcsec (- (D^{2} - 3 D + 2) y) + 2 πn) : x = - ln (arcsec (- y (D^{2} - 3 D + 2)) + 2 πn)

- x = ln (arcsec (- (D^{2} - 3 D + 2) y) + 2 πn)

Divide both sides by

- 1

- x = ln (arcsec (- (D^{2} - 3 D + 2) y) + 2 πn)

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{ln ( arcsec ( - ( D ^{2} - 3 D + 2 ) y ) + 2 πn )}{- 1}

Simplify

x = - ln (arcsec (- y (D^{2} - 3 D + 2)) + 2 πn)

x = - ln (arcsec (- y (D^{2} - 3 D + 2)) + 2 πn)

x = - ln (arcsec (- y (D^{2} - 3 D + 2)) + 2 πn)

Solve

e^{- x} = - arcsec (- (D^{2} - 3 D + 2) y) + 2 πn : x = - ln (- arcsec (- y (D^{2} - 3 D + 2)) + 2 πn)

e^{- x} = - arcsec (- (D^{2} - 3 D + 2) y) + 2 πn

Apply exponent rules

e^{- x} = - arcsec (- (D^{2} - 3 D + 2) y) + 2 πn

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{- x}) = ln (- arcsec (- (D^{2} - 3 D + 2) y) + 2 πn)

Apply log rule:

ln (e^{a}) = a

ln (e^{- x}) = - x

- x = ln (- arcsec (- (D^{2} - 3 D + 2) y) + 2 πn)

- x = ln (- arcsec (- (D^{2} - 3 D + 2) y) + 2 πn)

Solve

- x = ln (- arcsec (- (D^{2} - 3 D + 2) y) + 2 πn) : x = - ln (- arcsec (- y (D^{2} - 3 D + 2)) + 2 πn)

- x = ln (- arcsec (- (D^{2} - 3 D + 2) y) + 2 πn)

Divide both sides by

- 1

- x = ln (- arcsec (- (D^{2} - 3 D + 2) y) + 2 πn)

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{ln ( - arcsec ( - ( D ^{2} - 3 D + 2 ) y ) + 2 πn )}{- 1}

Simplify

x = - ln (- arcsec (- y (D^{2} - 3 D + 2)) + 2 πn)

x = - ln (- arcsec (- y (D^{2} - 3 D + 2)) + 2 πn)

x = - ln (- arcsec (- y (D^{2} - 3 D + 2)) + 2 πn)

x = - ln (arcsec (- y (D^{2} - 3 D + 2)) + 2 πn), x = - ln (- arcsec (- y (D^{2} - 3 D + 2)) + 2 πn)

Combine all the solutions

x = - ln (arcsec (y (D^{2} - 3 D + 2)) + 2 πn), x = - ln (- arcsec (y (D^{2} - 3 D + 2)) + 2 πn), x = - ln (arcsec (- y (D^{2} - 3 D + 2)) + 2 πn), x = - ln (- arcsec (- y (D^{2} - 3 D + 2)) + 2 πn)

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor x,(D^2-3D+2)y=sec^2(e^{-x}) ?
The general solution for solvefor x,(D^2-3D+2)y=sec^2(e^{-x}) is