Solve x^2 times second derivative -3x times first derivative -5y =sin(logx) my solution is integrating both sides we have sinxlogx
Your school subject is "St. Eugene university"? That must be very interesting.
First of all: St Eugene university is not a school subject
second: I can't make any sense out of your post.
Is it:
x^2 * (second derivative of -3x) * (first derivative of -5y) = sin(logx) ??
is it the derivative with respect to x ?
then 2nd derivative of -3x is 0, and
you have
x^2 * (0) * (first derivative of -5y) = sin(logx)
0 = sin(logx)
then logx = 0, not possible
or logx = π , x = appr 1385.46
(check: log 1385.46 = 3.14159... )
or logx = 2π , x = 1,919,487.58..
or logx = 3π , x = getting silly
I interpret the somewhat garbled language as
x^2 y" - 3xy' - 5y = sin(logx)
well, the homogeneous equation has solution
y = ax^5 + b/x
Now apply the methods for the Euler equation to get the desired solution.
To solve the given differential equation, we'll follow the steps:
Step 1: Recognize the given equation.
The given equation is a second-order linear homogeneous ordinary differential equation.
It can be written in the following form:
x^2(d^2y/dx^2) - 3x(dy/dx) - 5y = sin(logx)
Step 2: Find the general solution.
To find the general solution, we'll assume a solution of the form y = x^r, where r is a constant to be determined.
We'll start by finding the first and second derivatives of y:
dy/dx = rx^(r-1)
d^2y/dx^2 = r(r-1)x^(r-2)
Substituting these derivatives into the differential equation, we get:
x^2 * r(r-1)x^(r-2) - 3x * rx^(r-1) - 5x^r = sin(logx)
Simplifying the equation, we have:
r(r-1)x^r - 3rx^r - 5x^r = sin(logx)
Dividing through by x^r, we get:
r(r-1) - 3r - 5 = (sin(logx)) / (x^r)
Step 3: Solve the auxiliary equation.
To solve the auxiliary equation, we'll set the left side of the equation equal to zero:
r(r-1) - 3r - 5 = 0
Expanding and rearranging the equation, we have:
r^2 - r - 3r + 3 - 5 = 0
r^2 - 4r - 2 = 0
Solving this quadratic equation, we can use the quadratic formula:
r = (-(-4) ± √((-4)^2 - 4(1)(-2))) / (2(1))
r = (4 ± √(16 + 8)) / 2
r = (4 ± √24) / 2
r = (4 ± 2√6) / 2
r = 2 ± √6
So, we have two roots for the auxiliary equation: r₁ = 2 + √6 and r₂ = 2 - √6.
Step 4: Write the general solution.
The general solution to the given differential equation is given by the linear combination of the two solutions, y₁ and y₂, corresponding to the roots of the auxiliary equation.
Using the assumed form y = x^r, we can obtain the two solutions:
y₁ = x^(2 + √6)
y₂ = x^(2 - √6)
Therefore, the general solution is:
y = C₁ * x^(2 + √6) + C₂ * x^(2 - √6),
where C₁ and C₂ are arbitrary constants.
Step 5: Verify the answer.
To verify the solution, substitute y back into the original differential equation:
x^2(d^2y/dx^2) - 3x(dy/dx) - 5y = sin(logx)
Substituting y = C₁ * x^(2 + √6) + C₂ * x^(2 - √6), and taking the necessary derivatives, we can check if the equation holds true for any choice of constants C₁ and C₂.