Solve the differential equation by variation of parameters. 1 2 + ex y" + 3y + 2y =

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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provide complete solution showing the W1, W2, U'1, U'2, U1, U2, y(x)

Solve the differential equation by variation of parameters.
1
2 + ex
y" + 3y' + 2y =
Transcribed Image Text:Solve the differential equation by variation of parameters. 1 2 + ex y" + 3y' + 2y =
Step 5
We have found the following derivatives of the functions u₁(x) and u₂(x).
U1
42
42
Find the following integrals. Do not add any constants of integration, as the resulting functions are used to find just one particular solution.
-12 + x exox
-dx
=
+
= In et + 2
- 1-2 +0
e2x
=
ex
(2 + e*)
e2x
(2 + e*)
-dx
ex
=
= 2 ln et + 2
y(x): =
Step 6
We have found the following complementary function for the given differential equation.
Yc=c₁e + c₂e
We have also found that for y₁ = ex, , Y2 = e 4₁ =
2x
To finish, use the fact that y = Yc +
is the general solution of the nonhomogeneous differential equation to solve.
р
-2x
+2e
et +2 +eIn et + 2
-X
C₁e + C₂
In(2 + ex), and u₂ = 2 In(2 + e*) - e, a particular solution for the equation is given by Yp = U₁Y₁ + ₂Y 2.
Transcribed Image Text:Step 5 We have found the following derivatives of the functions u₁(x) and u₂(x). U1 42 42 Find the following integrals. Do not add any constants of integration, as the resulting functions are used to find just one particular solution. -12 + x exox -dx = + = In et + 2 - 1-2 +0 e2x = ex (2 + e*) e2x (2 + e*) -dx ex = = 2 ln et + 2 y(x): = Step 6 We have found the following complementary function for the given differential equation. Yc=c₁e + c₂e We have also found that for y₁ = ex, , Y2 = e 4₁ = 2x To finish, use the fact that y = Yc + is the general solution of the nonhomogeneous differential equation to solve. р -2x +2e et +2 +eIn et + 2 -X C₁e + C₂ In(2 + ex), and u₂ = 2 In(2 + e*) - e, a particular solution for the equation is given by Yp = U₁Y₁ + ₂Y 2.
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