1. The given family of functions is the general solution of the differential equ

Question

1. The given family of functions is the general solution of the differential

equation on the indicated interval. Find a member of the family that is a solution of the

initial-value problem.??=?? ??+?? ????????, (0, +?)

??2?????????? +??=0, ??(1)=3, ???(1)=?1

3.Find the general solution of the given second-order differential equation.

4???? + 20??? + 25?? = 0

4. Find the general solution of the given second-order differential equation.

2????+ 2???+??= 0

5. Solve the differential equation by variation of parameters.

???????=cosh??, where cosh??=????+????? 2

6. Solve the given differential equation by undetermined coefficients.

???? +3??? +2??=4cos??+4sin??

7. Solve the Cauchy-Euler equation:.

??2???? ?3????? +13?? = 4+3??

8. Find the general solution of the given higher-order differential equation.

?????+3?????4????12??=6

9. Use the definition to find Laplace Transforms , where. ??(??) = 4?? ? 10

10. Use the linear property of the Inverse Laplace Transforms to find the given inverse

Laplace transform.

???1((s+4)^2/s^4)

Explanation / Answer

rules are one quesiton per post I answered three

1) y' = c + c ln x + c = 2 c + c ln x
y'' = c/x
so
x^2 y'' - x y' + y = 0
cx - 2cx + c xln x + cx + cx ln x = 0
so satisfies

no apply intiial condition
y(1) = c = 3
so c = 3
y = 3x + 3 x ln x
2) characteristic equations
4 c^2 + 20 c + 25 = 0
(2c+5)^2 = 0
c = -5/2
since only one root
so y = A e^(-5/2 x) + B x e^(-5/2x)
3) characterist equation
2 c^2 + 2c + 1= 0
c=-1/2 -i/2 or -1/2 + i/2
so y = A e^((-1/2 -i/2)x) + Be^((-1/2 +i/2)x)

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