Finding the general solution of the equation y '' + 4Y '- 5Y = 0

Finding the general solution of the equation y '' + 4Y '- 5Y = 0

The characteristic equation is p * P + 4p-5 = 0
The solution is p = 1, P = - 5
y=C1 e^x + C2 e^-5x

Finding the general solution of the equation y '' - 4Y '+ 13 = 0

Eigenvalue = 0,4, corresponding to homogeneous general solution: a + be ^ (4x)
Special solution: y = - 13 / (d (D-4)) = - 13 (- 1 / 4-D / 16) / D = 13X / 4
D is a differential operator
In fact, it can be observed by y = 13X / 4
General solution y = a + be ^ (4x) + 13X / 4

The general solution of the equation y "- 4Y + 13 = 0 is

General solution of differential equation y '' - 4Y + 13 = 0
The characteristic equation corresponding to the homogeneous equation y '- 4Y = 0 is R & # 178; - 4 = 0, so R & # 8321; = - 2, R & # 8322; = 2
So the general solution of the homogeneous equation is y = C &; e ^ (- 2x) + C &; e ^ (2x)
Let's find a special solution y *;
According to the structure of the original problem, countable special solution is y * = a, so y * '= 0; y *' = 0; y * '= 0;
Substituting into the original formula, we get - 4A + 13 = 0, so a = 13 / 4;
So the general solution of the original equation is y = C &; e ^ (- 2x) + C &; e ^ (2x) + 13 / 4

If 3mx ^ 2Y is a quintic monomial with a coefficient of 1, then M =, n=

If 3mx ^ 2Y ^ n is a quintic monomial with coefficient 1 of X, y,
Then 3M = 1,2 + n = 5
∴m=1/3,n=3

What is the number of monomials?

The degree of AB is 1, AB + B = 40 is a quadratic equation
There are two unknowns,
The square of a is called twice
A is once. Here's an example

The number of monomials

It is the sum of the times of all the letters in the formula, such as the binomial ABCD. Because the exponent of each letter (that is, the times) is 1, the times of this binomial is 4

What is called several monomials
For example, π R & # 178; is a binomial of several times - ABC is a binomial of several times

For example, π R & # 178; is a binomial - ABC is a binomial,
The sum of the exponent of each factor letter is the number

If - (M / 4) (x ＾ m-1) y ＾ 2n is a quintic monomial with a coefficient of-1, the value of m.n is obtained

The coefficient is - 1, so - M / 4 = - 1, M = 4
So the number of times of X is three, and the number of times of monomials is five, so the number of times of Y is two
We get n = 1
So m = 4, n = 1

It is known that the sum of NX ^ 3Y ^ m and MX ^ n-1y are monomials, and the sum of these two monomials can be obtained

n-1=3,m=1
n=4,m=1
nx^3y^m+mx^n-1y
=5x^3y

If the sum of the monomial MX ^ n + 1y ^ 2m + 5 and x ^ 3Y is the monomial M-N=

n+1=3
2m+5=1
∴m=-2
n=2
∴m-n=-2-2=-4