Solve the given differential equation by undetermined coefficients y''+7y'+6y=30

Question

Solve the given differential equation by undetermined coefficients

y^{″} + 7 y^{'} + 6 y = 30

Answer 1

Consider the differential equation

y^{″} + 7 y^{'} + 6 y = 30

...(1)
Rewrite the equation,

(D^{2} + 7 D + 6) y = 30

where

D = \frac{d}{d x}

Auxiliary equation is,

f (m) = 0

m^{3} + 7 m + 6 = 0

m^{2} + 6 m + m + 6 = 0

m (m + 6) + 1 (m + 6) = 0

(m + 6) (m + 1) = 0

m = - 1, - 6

Hence the complementary solution is,

y_{c} = c_{1} e^{- x} + c_{2} e^{- 6 x}

Answer 2

Medicim6

Answered 2021-12-18 Author has 33 answers

Can I get full answer, please. I think tha is not full.

This is helpful 0

Answer 3

Oh, sure, sorry
Apply the undetemined coefficient method to find the particular solution as follows:
$y_{p} = A x + B$
Find the first and second derivatives,
$y_{p}^{'} = A,$
$y_{p}^{″} = 0$
Substitute these values in equation (1)
$0 + 7 A + 6 (A x + B) = 30$
$(7 A + 6 B) + 6 A x = 30$
Compare the coeficients of constant and x terms,
$6 A = 0$
$7 A + 7 B = 0$
Solve the above two equations,
$A = 0 a n d B = 5 (A = 0), (7 (0) + 6 B = 30), (6 B = 30), (B = 5) :$
Hence the particular solution is,
$y_{p} = 0 (x) + 5$
$= 5$
The general solution is,
$y = y_{c} + y_{p}$
$y_{c} = c_{1} e^{- x} + c_{2} e^{- 6 x} + 5$
Therefore, the required general solution is $y_{c} = c_{1} e^{- x} + c_{2} e^{- 6 x}$

Solve the given differential equation by undetermined coefficients y''+7y'+6y=30

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