cpyles1 said:I've been trying to solve this one for a while. My professor wasn't even sure how to do it. Any suggestions?
y''+t*y+y*y'=sin(t)
Remove[a];
nmax = 25;
myleftside = Sum[n*(n - 1)*Subscript[a, n]*t^(n - 2), {n, 0, nmax}] +
Sum[Subscript[a, n - 3]*t^(n - 2), {n, 3, nmax + 2}] +
Sum[Subscript[a, k]*Subscript[a, n - k]*(n - k)*t^(n - 1), {n, 0, nmax + 1}, {k, 0, n}];
myrightside = Sum[((-1)^n*t^(2*n + 1))/(2*n + 1)!, {n, 0, nmax}];
myclist = Flatten[Table[Solve[Coefficient[myleftside, t, n] == Coefficient[myrightside, t, n],
Subscript[a, n + 2]], {n, 0, nmax}]];
Subscript[a, 0] = 0;
Subscript[a, 1] = 1;
mysec = Table[Subscript[a, n] = Subscript[a, n] /. myclist, {n, 2, nmax}];
thef[t_] := Sum[Subscript[a, n]*t^n, {n, 0, nmax}];
p1 = Plot[thef[t], {t, 0, 2}, PlotStyle -> Red];
mysol = NDSolve[{Derivative[2][y][t] + t*y[t] + y[t]*Derivative[1][y][t] == Sin[t], y[0] == 0,
Derivative[1][y][0] == 1}, y, {t, 0, 2}];
p2 = Plot[y[t] /. mysol, {t, 0, 2}, PlotStyle -> Blue];
Show[{p1, p2}]A second order, nonlinear differential equation is a mathematical equation that involves the second derivative of a function, as well as the function itself, and possibly other terms. Nonlinear refers to the fact that the equation is not a linear function, meaning the variables are not directly proportional to each other.
Solving a second order, nonlinear differential equation typically involves using techniques such as substitution, separation of variables, or transforming the equation into a linear one. Depending on the complexity of the equation, it may also require numerical methods or computer software to find a solution.
Second order, nonlinear differential equations are commonly used in science to model complex systems and phenomena, such as chemical reactions, population growth, and fluid dynamics. They allow for a more accurate representation of real-world situations and can help predict future behavior.
One example of a second order, nonlinear differential equation is the Lotka-Volterra model, which describes the predator-prey relationship in a biological ecosystem. It involves the second derivative of both the predator and prey populations, as well as other nonlinear terms such as the interaction between the two populations.
Second order, nonlinear differential equations are used in a wide range of fields, including physics, biology, chemistry, engineering, and economics. They can be used to model systems and phenomena such as oscillations, growth and decay, vibrations, and electrical circuits.