The question at hand is a step in solving the manifolds of a non-linear dynamical system. However the part I need help with is a step using the integrating factor method. We need to solve the 1st order ODE $$\frac{dy}{dt}-y=(x_0)^2e^{-2t}.$$ Me and my lecturer both agreed that the integration factor is $e^{-t}$, and working it out the way I usually do I got my answer: $$e^{-t}y(t)=\frac{{x_0}^2}{3}-\frac{{x_0}^2}{3}e^{-3t}.$$ However this is apparently wrong as my lecturer had an extra term, $y_0$, which I'm assuming is an integration constant but I cannot work out where it has come from. So his answer was $$e^{-t}y(t)=y_0+\int^{t}_{0}x_0^2e^{-3s}ds=y_0+\frac{{x_0}^2}{3}-\frac{{x_0}^2}{3}e^{-3t}.$$ It then goes on to have much importance in calculating the manifold so please can someone tell me how he obtained the $y_o$!
2025-01-13 17:10:21.1736788221
Integration factor method with constants of integration
166 Views Asked by PokerF4ce https://math.techqa.club/user/pokerf4ce/detail At
1
There are 1 best solutions below
Related Questions in ORDINARY-DIFFERENTIAL-EQUATIONS
- General solution to a system of differential equations
- ODE existence of specific solutions
- How to Integrate the Differential Equation for the Pendulum Problem
- Question about phase portrait and invariant subspaces
- Help in Solving a linear Partial differential equation
- Elimination of quantifiers in the strucure of polynomials and in the structure of exponentials
- Verifying general solution to differential equation
- Integrating $ \frac{\mathrm{d}^{2}v}{\mathrm{d}y^{2}} = \frac{\mathrm{d}p}{\mathrm{d}x} $
- Solving differential equation and obtain expressions for unknowns?
- For what value of $k$ is $2e^{4x}-5e^{10x}$ a solution to $y''-ky'+40y=0$?
Related Questions in INTEGRATING-FACTOR
- Exact Equations and Integrating Factor
- Solve $y''+2y'-8y=e^{2x}$ using integrating factor method
- Proof dividing two integrating factors = k is solution to an exact equation
- second order exact differential equation
- Determining integrating factor
- Common integrating factor of two differential equations.
- Transforming a partial derivative into a total derivative via integrating factor
- Why is this first-order ODE amenable to the method of integrating factor/separable?
- O.D.E Integrating Factor Help
- How should I approach this ivp problem with two differentiation about x?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Refuting the Anti-Cantor Cranks
- Find $E[XY|Y+Z=1 ]$
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- What are the Implications of having VΩ as a model for a theory?
- How do we know that the number $1$ is not equal to the number $-1$?
- Defining a Galois Field based on primitive element versus polynomial?
- Is computer science a branch of mathematics?
- Can't find the relationship between two columns of numbers. Please Help
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
- Alternative way of expressing a quantied statement with "Some"
Popular # Hahtags
real-analysis
calculus
linear-algebra
probability
abstract-algebra
integration
sequences-and-series
combinatorics
general-topology
matrices
functional-analysis
complex-analysis
geometry
group-theory
algebra-precalculus
probability-theory
ordinary-differential-equations
limits
analysis
number-theory
measure-theory
elementary-number-theory
statistics
multivariable-calculus
functions
derivatives
discrete-mathematics
differential-geometry
inequality
trigonometry
Popular Questions
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- How to find mean and median from histogram
- Difference between "≈", "≃", and "≅"
- Easy way of memorizing values of sine, cosine, and tangent
- How to calculate the intersection of two planes?
- What does "∈" mean?
- If you roll a fair six sided die twice, what's the probability that you get the same number both times?
- Probability of getting exactly 2 heads in 3 coins tossed with order not important?
- Fourier transform for dummies
- Limit of $(1+ x/n)^n$ when $n$ tends to infinity
Your differential equation is $$\frac{dy}{dt}-y=x_0^2e^{-2t}\tag1$$ Clearly, it is a first order linear differential equation and the integrating factor is $$\text{I.F.}~~=~e^{\int (-1)~dt}=e^{-t}$$ Now multiplying both side of $(1)$ by I.F. we have, $$e^{-t}\frac{dy}{dt}-ye^{-t}=x_0^2e^{-3t}$$ $$\implies \frac{d}{dt}\left(ye^{-t}\right)=x_0^2e^{-3t}$$ $$\implies d\left(ye^{-t}\right)=x_0^2e^{-3t}dt\tag2$$ Integrating equation $(2)$ between the limit $~0~$ to $~t~$, we have $$\left[ye^{-t}\right]_0^t=\int_0^t x_0^2e^{-3s}ds$$ $$\implies y(t)e^{-t}-y(0)e^{-0}=\int_0^t x_0^2e^{-3s}ds$$ $$\implies y(t)e^{-t}-y(0)=\int_0^t x_0^2e^{-3s}ds$$ $$\implies y(t)e^{-t}-y_0=\int_0^t x_0^2e^{-3s}ds$$ $$\implies y(t)e^{-t}=y_0+\int_0^t x_0^2e^{-3s}ds\tag3$$ $$\implies y(t)e^{-t}=y_0-\dfrac 13 x_0^2\left[e^{-3s}\right]_0^t$$ $$\implies y(t)e^{-t}=y_0-\dfrac 13 x_0^2\left[e^{-3t}-1\right]$$ $$\implies y(t)e^{-t}=y_0+\dfrac {x_0^2}{3}-\dfrac {x_0^2}{3} e^{-3t}\tag4$$ Combining equation $(3)$ and $(4)$ we have $$y(t)e^{-t}=y_0+\int^{t}_{0}x_0^2e^{-3s}ds=y_0+\frac{{x_0}^2}{3}-\frac{{x_0}^2}{3}e^{-3t}.$$