Solve #23, the initial-value problem using whatever method is appropriate. 19. y + y sin x = 0,y(t) = 1/220. ty=t²e t,y(1) = 321. (y+e)dy+ (e * x) dx = 0, y(0)22. (x²+1)y + 2x³y = 6xe x² y(0) =23. tx dx =t² + 3x², x(1):dtdt24. (y + 2x sin y cos y) y==12= 3x² sin² y,

Answered: 23. tx d = t² + 3x², x(1) = 1 dx dt

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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Question

Solve #23, the initial-value problem using whatever method is appropriate.

19. y + y sin x = 0,
y(t) = 1/2
20. ty=t²
e t,
y(1) = 3
21. (y+e)dy
+ (e * x) dx = 0, y(0)
22. (x²+1)y + 2x³y = 6xe x² y(0) =
23. tx dx =
t² + 3x², x(1):
dt
dt
24. (y + 2x sin y cos y) y
=
=1
2
= 3x² sin² y,

Expert Solution

Step 1

Given:

$t x \frac{d x}{d t} = t^{2} + 3 x^{2}$ and $x (1) = 1$

To find:

The solution to the given initial value problem.

Concept used:

The differential equation $\frac{d y}{d x} = f (x, y)$ is said to be homogeneous differential equations only if $f (λ x, λ y) = λ^{n} f (x, y)$ , where $λ$ is any constants and $n \in ℤ$ .

Formula used:

i) $\frac{d}{d x} (f (x) g (x)) = f (x) \frac{d}{d x} (g (x)) + g (x) \frac{d}{d x} (f (x))$

ii) $\int \frac{f' (x)}{f (x)} d x = \ln |f (x)| + c$ , where $c$ is the integrating constant.

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