Question
upstudy study bank question image url

Example2: \( y^{\prime \prime}+y^{\prime}-2 y=x e^{x} \)

Ask by Bird Stewart. in Iraq
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

الحل العام للمعادلة التفاضلية هو: \[ y = C_1 e^{-2x} + C_2 e^{x} + e^{x}\left(\frac{1}{6}x^2 - \frac{1}{9}x\right) \] حيث \( C_1 \) و \( C_2 \) ثوابت تعيينها بناءً على الشروط الابتدائية.

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Extra Insights

To solve the differential equation \( y^{\prime \prime}+y^{\prime}-2 y=x e^{x} \), we start by finding the complementary solution (homogeneous part) by solving \( y^{\prime \prime} + y^{\prime} - 2y = 0 \). The characteristic equation is \( r^2 + r - 2 = 0 \), which factors into \( (r-1)(r+2) = 0 \), yielding roots \( r = 1 \) and \( r = -2 \). Thus, the complementary solution is \( y_c = C_1 e^{x} + C_2 e^{-2x} \). Next, for the particular solution \( y_p \) for the non-homogeneous part \( x e^{x} \), we can use the method of undetermined coefficients. Guess a solution of the form \( y_p = (Ax + B)e^{x} \). Upon differentiating and substituting into the original equation, we solve for the coefficients \( A \) and \( B \). By combining the complementary and particular solutions, we find the general solution of the differential equation. Now, let’s get into some exciting facts about it! This equation is a second-order linear ordinary differential equation (ODE), and its solutions can describe phenomena in physics and engineering, like vibrations, waves, or thermal systems. In control theory, similar equations govern the behavior of control systems, allowing engineers to design systems that respond predictively to inputs. Additionally, the method of solving such ODEs with the superposition principle and undetermined coefficients is a common theme in differential equations. If you're interested in diving deeper, look into differential equations textbooks where theories are well-explored, or check online courses that provide problem sets. Places like MIT OpenCourseWare offer fantastic resources!

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy