Solving Linear ODE: dy/dx-2y=x^2e^2x

  • Thread starter NewtonianAlch
  • Start date
  • Tags
    Linear Ode
In summary, the given equation is solved using an integrating factor of e^{2x}. After multiplying both sides by the integrating factor and simplifying, the left-hand side is contracted using the chain rule. Integrate both sides and solve for y, the final solution is y = \frac{1}{3}e^{2x}x^{3}+Ce^{2x}.
  • #1
NewtonianAlch
453
0

Homework Statement


Solve [itex]\frac{dy}{dx}[/itex] - 2y = x[itex]^{2}[/itex]e[itex]^{2x}[/itex]

The Attempt at a Solution



Integrating factor = e[itex]^{2x}[/itex]

So we multiply through the given equation by the integrating factor and get:

e[itex]^{2x}[/itex][itex]\frac{dy}{dx}[/itex] - 2e[itex]^{2x}[/itex]y = x[itex]^{2}[/itex]e[itex]^{4x}[/itex]

Contract the left-hand side via the chain rule to get:

[itex]\frac{d}{dx}[/itex](e[itex]^{2x}[/itex]y) = x[itex]^{2}[/itex]e[itex]^{4x}[/itex]

Integrate both sides

e[itex]^{2x}[/itex]y = [itex]\frac{1}{32}e^{4x}[/itex](8x[itex]^{2}[/itex]-4x+1)+C

Now divide through by e[itex]^{2x}[/itex] and the equation definitely does not equal what Wolfram Alpha gives as the solution:

y = [itex]\frac{1}{3}[/itex]e[itex]^{2x}[/itex]x[itex]^{3}[/itex]+Ce[itex]^{2x}[/itex]

I checked some of the parts individually with Wolfram, such as the integration of the right-hand side and that was correct, so I'm not too sure what's causing the difference in answers.
 
Physics news on Phys.org
  • #2
Grr, nvm. I forgot the minus sign in the integrating factor -.-
 

FAQ: Solving Linear ODE: dy/dx-2y=x^2e^2x

What is a linear ODE?

A linear ODE (ordinary differential equation) is an equation that involves a function and its derivatives in a linear way. This means that the function and its derivatives appear to the first power and are not multiplied or divided by each other.

How do you solve a linear ODE?

To solve a linear ODE, you need to rearrange the equation so that it is in the standard form of dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x. Then, you can use an integrating factor to solve the equation and find the solution for the function y(x).

What is the role of the integrating factor in solving linear ODEs?

The integrating factor is a function that is multiplied to both sides of the linear ODE to transform it into a form that can be easily integrated. It is typically determined by the coefficient of y in the standard form of the equation.

How do you apply the integrating factor to solve the given equation?

In the equation dy/dx - 2y = x^2e^2x, the integrating factor is e^(-2x). You would multiply both sides of the equation by this integrating factor to get e^(-2x)dy/dx - 2e^(-2x)y = x^2. Then, you can integrate both sides and solve for y(x).

Can you use a computer program to solve linear ODEs?

Yes, there are many computer programs and software packages that can solve linear ODEs using numerical methods. These methods involve approximating the solution using a series of points and can be very accurate. However, it is still important to understand the concepts and techniques behind solving linear ODEs by hand.

Similar threads

Solve the given problem involving integration
Replies
6
Views
440
Verify Green's Theorem in the given problem
Replies
10
Views
1K
First Order Diffy Q Problem with Bernoulli/Integrating Factors
Replies
4
Views
1K
Unsolvable max/min of surface with constraint
Replies
9
Views
1K
Solve the given first order differential equation
Replies
8
Views
1K
Solving an Euler differential equation
Replies
6
Views
2K
Evaluate ##\int_{-\infty}^{\infty} e^{-|x|}\delta(x^2 +2x -3) dx##
Replies
7
Views
1K
Question on Two Differentiation Techniques
Replies
25
Views
1K
Solve this problem that involves implicit differentiation
Replies
15
Views
1K
Find the length of the curve C
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top