Integral of exponential over polynomial

In summary, the conversation discusses the process of solving a variable coefficient ODE using the method of reduction of order. The approach involves assuming a solution in the form of y=c1*y1+c2*y2, with y1=x and y2 being the second solution reached through integration. The integral for the second solution involves using integration by parts and results in an expression that is similar to the error function.
  • #1
AAO
17
2

Homework Statement


Solve the ODE: y''+x*y'-y=0

Homework Equations

The Attempt at a Solution


Since this is a variable coefficient ODE, I have used the method of reduction of order, and assumed the solution in the form: y=c1*y1+c2*y2

In this case: y1=x, and I have the reached the integral below for the second solution (y2), Can anyone tell me how to approach this integral:

v=integral[ (x^-2) * exp (-0.5*x^2) dx]
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
AAO said:

Homework Statement


Solve the ODE: y''+x*y'-y=0

Homework Equations

The Attempt at a Solution


Since this is a variable coefficient ODE, I have used the method of reduction of order, and assumed the solution in the form: y=c1*y1+c2*y2

In this case: y1=x, and I have the reached the integral below for the second solution (y2), Can anyone tell me how to approach this integral:

v=integral[ (x^-2) * exp (-0.5*x^2) dx]
Integration by parts leaves you with an integral that is (up to some constants) the error function.
 
Last edited by a moderator:

FAQ: Integral of exponential over polynomial

What is the integral of exponential over polynomial?

The integral of exponential over polynomial is a mathematical expression that represents the area under the curve of an exponential function divided by a polynomial function. It can be written as ∫e^x / (ax^n + bx^(n-1) + ... + cx + d) dx.

Is it possible to solve the integral of exponential over polynomial analytically?

Yes, it is possible to solve the integral of exponential over polynomial analytically by using integration techniques such as substitution, integration by parts, or partial fractions.

What is the significance of the integral of exponential over polynomial in real-world applications?

The integral of exponential over polynomial is commonly used in physics and engineering to solve problems involving exponential growth or decay combined with polynomial functions. It is also used in statistics to calculate the expected value of certain probability distributions.

Are there any special cases or restrictions when solving the integral of exponential over polynomial?

Yes, the integral of exponential over polynomial can only be solved when the degree of the polynomial is greater than the power of the exponential function. Additionally, the polynomial must not have repeated roots or a zero degree term.

Can numerical methods be used to approximate the integral of exponential over polynomial?

Yes, when the integral of exponential over polynomial cannot be solved analytically, numerical methods such as the trapezoidal rule or Simpson's rule can be used to approximate the value of the integral.

Similar threads

Variation of parameter VS Undetermined Coefficients
Replies
7
Views
1K
Calculate this Integral around the Circular Path using Green's Theorem
Replies
3
Views
1K
Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##
Replies
7
Views
1K
How Do I Solve a Second Order ODE with Non-Constant Coefficients?
Replies
2
Views
1K
Solve for the solution of the differential equation
Replies
9
Views
2K
Abel's Equation and Wronskian for find out y2
Replies
1
Views
2K
Solve p = P(2X <= Y^2) using double integral
Replies
4
Views
1K
The General Solution of the DE y''+y'=tan(t) ?
Replies
2
Views
2K
How to check if a polynomial is irreducible over the rationals
Replies
18
Views
3K
Find constants that satisfy integrals?
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top