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First Order Differential Equations, Partial DE's/ Linear Dependence/Independance

1. Find solutions to the given Cauchy- Euler equation
(a) xy'+ y =0 (b) x2y'' + xy'+y =0 ; y(1) =1, y'(1) =0

2. Find a solution to the initial value problem
x2y' + 2xy = 0; y (1) = 2

3. Find the general solution to the given problems
(a) Y' + (cot x)y = 2cosx (b) (x-5)(xy'+3y) = 2

4. Solve the Bernoulli equation y' = xy3 - 4y

5. Use separation of variables to solve the verhulst population problem

N' (t) = (a-bN) N, N (0) = N0; a,b > 0

6. Verify that each of the given functions is a solution of the given differential equation, and then use the Wronskian to determine linear dependence/ independence
Y''' - y''- 2y' = 0 {1, e-x, e2x}

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Solution Preview

1a.
The equation is:
(1.1)
We "guess" a solution in the form:
(1.2)
We substitute this back in the equation:

(1.3)
Thus the solution to (1.1) is:
(1.4)
Where C is a constant to be determined from initial conditions.

1b.
The equation is:
(1.5)
We "guess" a solution in the form:
(1.6)
We substitute this back in the equation:

(1.7)
We have a single pure complex root with multiplicity of two Thus the solution to (1.5) is a linear combination
(1.8)
From initial conditions:


Thus:
(1.9)

2.
The equation is:
(1.10)
Note that we can write the left hand side as:
(1.11)
Thus, the equation is:
(1.12)
Integrating both ...

Solution Summary

The first order differential equations, partial DE's and linear functions are provided. The expert uses separation of variables to solve the verhulst population problems.

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