Avoid unpleasant integrals in solving IVP

In summary, the article discusses strategies for solving initial value problems (IVPs) while avoiding complex or unpleasant integrals. It emphasizes the use of analytical techniques and numerical methods that simplify the solution process, ensuring efficiency and accuracy without the need for challenging calculations. By employing appropriate transformations, approximations, or computational tools, one can effectively navigate around difficult integrals in IVP solutions.
  • #1
psie
266
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Homework Statement
Solve ##t^2y''-2ty'+2y=t^2\sin{t^4}, t>0## with initial values ##y(1)=2, y'(1)=5##.
Relevant Equations
See first two paragraphs below.
The formula I'm given is that the general solution to a linear inhomogeneous system ##x'(t)=A(t)x(t)+b(t)## is ##x(t)=F(t)\int F^{-1}(t)b(t)dt##, where ##F(t)## is the fundamental matrix to the linear homogenous system (here ##A(t)## is an ##n\times n## matrix function and ##b(t)## and ##n\times 1## matrix function, both continuous in some interval ##I\subset \mathbb R##).

Since a linear ##n##th order ODE ##y^{(n)}(t)+a_{n-1}(t)y^{(n-1)}(t)+\ldots +a_0(t)y(t)=f(t)## can be reduced to a system, the corresponding solution is ##y(t)=R_1 (t)\int K_n(t)f(t)dt##, where ##R_1(t)## is the first row of the fundamental matrix ##F(t)## and ##K_n(t)## the last column of the inverse of the fundamental matrix ##F^{-1}(t)##.

So here we are given the linear, second order ODE $$t^2y''-2ty'+2y=t^2\sin{t^4}.\tag1$$The homogeneous equation is a so-called Euler equation, i.e. of the form ##t^ny^{(n)}(t)+a_{n-1}t^{n-1}y^{(n-1)}(t)+\ldots+a_1ty'(t)+a_0y(t)=0##, where ##a_{n-1},\ldots,a_0## are constants (see Wikipedia). I will omit the details, but the general solution to the homogeneous equation of ##(1)## is $$y_h(t)=Ct+Dt^2.$$ From this we can construct the fundamental matrix and compute its inverse. It is $$F(t)=\begin{bmatrix}
t&t^2\\
1&2t
\end{bmatrix}\qquad F^{-1}(t)=\begin{bmatrix}
2/t&-1\\
-1/t^2&1/t
\end{bmatrix}.$$
So using the formula of the general solution to a linear ##n##th order ODE, i.e. ##y(t)=R_1 (t)\int K_n(t)f(t)dt##, we have $$y(t)=t\int (-\sin{t^4})dt+t^2\int \frac{\sin{t^4}}{t}dt.$$ We can define ##G(t)+C=\int (-\sin{t^4})dt## and ##H(t)+D=\int \frac{\sin{t^4}}{t}dt##, and we get $$y(t)=Ct+Dt^2+G(t)t+H(t)t^2.$$ But here I'm stuck, i.e. I do not know how check the initial values and find the solution to the IVP. Is there a way to avoid having to compute the integrals?
 
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  • #2
Do not use indefinite integrals here. Always fix a lower limit; the point at which the initial value is specified is convenient. This paritcular solution therefore vanishes at this point, leaving the coefficients of the homogenous solutions to satisfy the initial condition. I would therefore define [tex]\begin{split}
G(t) &= \int_1^t \sin u^4 \,du \\
H(t) &= \int_1^t \frac{\sin u^4}{u} \,du \end{split}[/tex] so that [itex]G(1) = H(1) = 0[/itex]. Then [tex]\begin{split}
y(x) &= Ct + Dt^2 - tG(t) + t^2H(t) \\
y'(x) &= C + 2Dt - G(t) - t\sin t^4 + 2tH(t) + t\sin t^4 \\
&= C + 2Dt - G(t) + 2tH(t).\end{split}[/tex]
 
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  • #3
Is there a formula (I would assume in terms of the fundamental matrix) for the particular solution of an ##n##th order linear ODE with any initial values?

Following my reasoning above, I would assume it would read $$y_p(t)=R_1 (t)\int_{t_0}^t K_n(u)f(u)du,$$ where ##R_1## is the first row of the fundamental matrix and ##K_n## the last column of the inverse of the fundamental matrix, but I'm not sure.
 
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  • #4
I'd be very grateful if someone could confirm the following.

In #1 I gave the solution to an inhomogeneous system, namely $$x(t)=F(t)\int (F(t))^{-1}b(t)dt.$$ Using definite integrals, i.e. we have some initial condition ##x(t_0)=x_0##, the above solution reads $$x(t)=F(t)(F(t_0))^{-1}x_0+F(t)\int_{t_0}^t (F(u))^{-1}b(u)du.$$ Now, the solution to the scalar ##n##th order linear ODE should just be the first component of ##x(t)##, meaning only the first component of ##F(t)(F(t_0))^{-1}x_0## and first component of ##F(t)\int_{t_0}^t (F(u))^{-1}b(u)du##. So the solution ##y(t)## of the scalar ##n##th order linear ODE, with initial values ##(y(t_0),\ldots,y^{(n-1)}(t_0))=x_0##, reads $$y(t)=P_1(t,t_0)x_0+R_1 (t)\int_{t_0}^t K_n(u)b(u)du,$$ where ##P_1(t,t_0)## is the first row of ##F(t)(F(t_0))^{-1}##, ##R_1(t)## the first row of ##F(t)## and ##K_n(u)## the ##n##th column of ##(F(u))^{-1}##.
 
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FAQ: Avoid unpleasant integrals in solving IVP

What are some common techniques to avoid unpleasant integrals in solving initial value problems (IVPs)?

Some common techniques include using Laplace transforms, employing numerical methods such as Euler's method or Runge-Kutta methods, and leveraging software tools like MATLAB or Mathematica that can handle complex integrations symbolically or numerically.

Can Laplace transforms always be used to avoid integrals in IVPs?

Laplace transforms are a powerful tool for solving linear differential equations with constant coefficients and can often simplify the process by converting the problem into an algebraic equation. However, they may not be suitable for non-linear differential equations or those with variable coefficients.

How do numerical methods help in avoiding unpleasant integrals?

Numerical methods approximate the solution of differential equations using discrete steps rather than solving them analytically. Techniques like Euler's method, Runge-Kutta methods, and finite difference methods can provide approximate solutions without requiring explicit integration.

What role do software tools play in solving IVPs without unpleasant integrals?

Software tools like MATLAB, Mathematica, and Python libraries (e.g., SciPy) can perform symbolic and numerical computations, allowing you to solve differential equations without manually performing complex integrations. These tools can handle a wide range of problems and provide visualizations of the solutions.

Are there any specific types of IVPs where avoiding integrals is particularly challenging?

Avoiding integrals can be particularly challenging in non-linear differential equations, equations with variable coefficients, and systems with intricate boundary conditions. In such cases, a combination of analytical and numerical methods or advanced mathematical techniques may be required.

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