Mathematics II 3) Linear Equation (first order first degree) Alinear of ordinary differential Equation (ODE)can always be put in to the standared from dy dx+p(x)y = Q(x) p = (integration factor) → p = e√ p(x)dx eqation[y = · = e−√ p(x)dx * (√ p * Q(x) + C)] Example// Solve = ex – Y - Solution/ dy +Y= ex -> dx + p(x)y = Q(x) dx y=ep(x)dx * = p(x)=1&Q(x) = ex p = e√ p(x)dx = ef1dx = ex * ( p + Q (x) + c ) = e × = ( √ e * + e ³dx + c) * * 1 ex ex + ( e²x dx + c) = ex (½ e²x + c) = ½e* + exc Exercise: Solve dy - 3y = x² dx 189
Mathematics II 3) Linear Equation (first order first degree) Alinear of ordinary differential Equation (ODE)can always be put in to the standared from dy dx+p(x)y = Q(x) p = (integration factor) → p = e√ p(x)dx eqation[y = · = e−√ p(x)dx * (√ p * Q(x) + C)] Example// Solve = ex – Y - Solution/ dy +Y= ex -> dx + p(x)y = Q(x) dx y=ep(x)dx * = p(x)=1&Q(x) = ex p = e√ p(x)dx = ef1dx = ex * ( p + Q (x) + c ) = e × = ( √ e * + e ³dx + c) * * 1 ex ex + ( e²x dx + c) = ex (½ e²x + c) = ½e* + exc Exercise: Solve dy - 3y = x² dx 189
Calculus: Early Transcendentals
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ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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Please (re)solve these examples in more detail as well..........
![Mathematics II
3) Linear Equation (first order first degree)
Alinear of ordinary differential Equation
(ODE)can always be put in to the standared from
dy
dx+p(x)y = Q(x)
p = (integration factor) → p = e√ p(x)dx
eqation[y =
· = e−√ p(x)dx * (√ p * Q(x) + C)]
Example// Solve = ex – Y
-
Solution/
dy
+Y= ex ->
dx
+ p(x)y = Q(x)
dx
y=ep(x)dx
*
=
p(x)=1&Q(x) = ex
p = e√ p(x)dx = ef1dx = ex
* ( p + Q (x) + c ) = e × = ( √ e * + e ³dx + c)
*
*
1
ex
ex + ( e²x dx + c) = ex (½ e²x + c) = ½e* + exc
Exercise: Solve dy - 3y = x²
dx
189](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1ecab767-ba67-485c-ab81-0a1604555596%2Fd60e75c0-5d2c-4c80-a3a3-b25090899df0%2Fzfoqrn8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Mathematics II
3) Linear Equation (first order first degree)
Alinear of ordinary differential Equation
(ODE)can always be put in to the standared from
dy
dx+p(x)y = Q(x)
p = (integration factor) → p = e√ p(x)dx
eqation[y =
· = e−√ p(x)dx * (√ p * Q(x) + C)]
Example// Solve = ex – Y
-
Solution/
dy
+Y= ex ->
dx
+ p(x)y = Q(x)
dx
y=ep(x)dx
*
=
p(x)=1&Q(x) = ex
p = e√ p(x)dx = ef1dx = ex
* ( p + Q (x) + c ) = e × = ( √ e * + e ³dx + c)
*
*
1
ex
ex + ( e²x dx + c) = ex (½ e²x + c) = ½e* + exc
Exercise: Solve dy - 3y = x²
dx
189
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