## Differential equations – Solution system of ODEs with additional parameters

I would like to solve one $$2 times 2$$ System of the form
$$frac {d} {d theta} T = TA, quad T (0) = Id$$
from where $$theta$$ is real and $$A$$ is of the form
$$A = begin {pmatrix} 0 & frac {e ^ {- i theta}} { lambda} \ frac {1} {36} e ^ {- i theta} left (9 Lambda + 2 ( lambda-1) ^ 2 (6 cos { theta} + cos {2 theta} + 6) right) & 0 end {pmatrix},$$
With $$lambda$$ a free parameter in the unit circle,

I am particularly interested in numerical solutions $$theta = 2 pi$$ depending on the additional parameters $$lambda$$, I'm pretty new with Mathematicaand I've tried it so far:

``````T[θ_] = {{T11[θ]T12[θ]}, {T21[θ]T22[θ]}};
ON[θ_] = {{0, E ^ (- Iθ) / λ}, {1/36 E ^ (- Iθ) (9λ + 2 (-1 + λ) ^ 2 (6 + 6 Cos[θ] + Cos[2 θ])), 0}};
sys = {T & # 39;[θ] == T[θ].ON[θ]};
``````

The previous code sets the system I want to solve, and now I'm trying to solve numerically. I tried it first

``````NSol = NDSolve[{sys, T11[0] == 1, T12[0] == 0, T21[0] == 0, T22[0] == 1}, {T11[θ]T12[θ]T21[θ]T22[θ]}, {θ}, {θ, 0, 2 Pi}];
``````

that gives me the output

``````NDSolve :: dupv: "Double variable θ found in NDSolve[<[<[<[<<1>>], "
``````

I tried too

``````Nsol2 = ParametricNDSolve[{sys, T11[0] == 1, T12[0] == 0, T21[0] == 0, T22[0] == 1}, {T11, T12, T21, T22}, {θ, 0, 2 Pi}, {λ}];
``````

what gives me as an issue $$T_ {11}, dots, T_ {22}$$ as Parametric Functions dependent on each other and on $$lambda$$,

I do not know if this is the right approach and if so, how to extract a numeric expression depending on it $$lambda$$ from the last issue – all I've seen in the documentation are examples that are recorded for specific parameter values. Any help is greatly appreciated.

## Differential Equations – Problem solving the first ODEs with DSolve

The problem I'm facing is the following. I am trying to solve a system of two coupled first order ODEs with DSolve:

``````eqns = {A * f & # 39;[x] == -x * f[x]/ 2 + B * g[x], A * g & # 39;[x] == x * g[x]/ 2 + B * f[x]};
DSolve[EQNS{f[EQNS{f[eqns{f[eqns{f[x]G[x]}, x]
``````

As output this only gives the same command that means from my collection that DSolve can not solve it.

But if I explicitly use the equations, it can suddenly be solved:

``````Fun = f[x] /. To solve[A*g&#39;[A*g'[A*g'[A*g'[x] == x * g[x]/ 2 + B * f[x]f[x]];
dfun = D[fun, x];
DSolve[A*f#39;[A*f'[A*f'[A*f'[x] == -x * f[x]/ 2 + B * g[x] /. {f[x] -> fun, f & # 39;[x] -> dfun},
G[x], x]
``````

Which gives

``````{{G[x] ->
C[2] ParabolicCylinderD[B^2/A, (I x)/Sqrt[A]]+
C[1] ParabolicCylinderD[(-A - B^2)/A, x/Sqrt[A]]}}
``````

That makes me think that I'm just doing something wrong on the first line, which is probably a stupid notation error. If someone could help me, I would be very grateful.

## Differential equations – How do you solve these ODEs with NDSolve?

The calculation changes `t = 0` there `sin[ψ`

``` eq1 = ω1 eq2 = ω2 and construct the linear combinations, eq1n = Simplify[eq1Sin[eq1Sin[eq1Sin[eq1Sin[ψ (* Cos[ψ (If this were not possible, the equations themselves could not be solved in principle.) Now replace eq1, eq2 by eq1n, eq2n, I1 = 2; I2 = 3; I3 = 4; s = NDSolveValue[{I1*ω1&#39;[{I1*ω1'[{I1*ω1'[{I1*ω1' I2 * ω2 & # 39; I3 * ω3 & # 39; eq1n == 0, eq2n == 0, ω3 ω1[0] == 2, ω2[0] == 3, ω3[0] == 4, ψ[0] == 0, φ[0] == 0, θ[0] == Pi / 6}, {ω1 plot[Evaluate@s[[1 ;; 3]], {t, 0, 120}, ImageSize -> Large]plot[Evaluate@s[[4 ;; 6]], {t, 0, 120}, ImageSize -> Large] Incidentally, the original equations can be solved by slightly changing the initial state ψ[0] == 0 to ψ[0] == 10 ^ -6, Another approach is to use the option. Method -> {"EquationSimplification" -> "Residual"} Everyone gives the same answer. ```
``` ```
``` Author AdminPosted on March 17, 2019Categories ArticlesTags differential, equations, NDSolve, ODEs, solve ```
``` How do I solve these Odes with NDSolve? I have six odes and can not use DSolve. So I tried NDSolve. But it does mean that there may be mistakes. The code looks like this: I1 = 2; I2 = 3; I3 = 4; NDSolve[{I1*ω1' I2 * ω2 & # 39; I3 * ω3 & # 39; 0, ω1[ t] == φ & # 39; sin[ψ[ψ[ψ[ψ t] == φ & # 39; cos[ψ[ψ[ψ[ψ t] == φ & # 39; 0] == 2, ω2[0] == 3, ω3[0] == 4, ψ[0] == 0, φ[0] == 0, θ[0] == Pi / 6}, {ω1, ω2, ω3, ψ, φ, θ}, {t, 0, 120}] I want to know how to avoid this mistake. Author AdminPosted on March 17, 2019Categories ArticlesTags NDSolve, ODEs, solve Classical analysis and Odes – Algebraic Riccati and WKB It is a one-liner to show that the algebraic Riccati equation (ARE) and the lowest-order form of WKB are the same for a linear ode. But I've looked everywhere on the Internet and it seems that this connection, despite its triviality is not recognized. It seems that there is a total separation between physicists who make many WKB and control people who do a lot of Riccati. Another trivial point is that a linear ode in psi has the Lie symmetry psi -> lambda psi, so the Riccati variable is the Lie invariant, and this leads to a reduction in order (or more precisely, a reduced one Order) followed by a quadrature.) Is this so obvious that it can never be mentioned? This finding means that not only linear equations are used, but every equation that is homogeneous in degree 1 and whose order is reduced by the Riccati transformation. Author AdminPosted on March 16, 2019Categories ArticlesTags algebraic, analysis, classical, ODEs, Riccati, WKB Classical analysis and odes – Failure of a Falconer distance problem in one dimension I am told that the Falconer distance guess in one dimension is trivial, but I really can not find any clue. I ask exactly the following question: For a compact set $$E subseteq mathbb R ^ n$$we define the set distance $$Delta (E) subseteq[0infty)[0infty)[0infty)[0infty)$$ his: $$Delta (E) = {| x-y |: x, y in E }.$$ Then, when $$n = 1$$do we ask the following questions? Can we find a compact set? $$E subseteq [0,1]$$ so that $$mathrm {dim} _H (E)> 1/2$$ but $$mathcal L ^ 1 ( Delta (E)) = 0$$? Can we find a compact set? $$E subseteq [0,1]$$ so that $$mathrm {dim} _H (E) = 1$$ but $$mathcal L ^ 1 ( Delta (E)) = 0$$? Author AdminPosted on March 7, 2019Categories ArticlesTags analysis, classical, dimension, distance, failure, Falconer, ODEs, problem Differential Equations – Simultaneous ODEs DSolve or NDSolve I try to solve Sim. ODEs as below. I try to use first DSolve which did not work and NDSolve which did not work. Can someone help? d = 1; r = 4; L = 7; W = 4; w = 2; b = 1; V[t_,s_]= Exp[(d+r)t]*Integrate[exp[(d+r)*(-x)] (s + d * v[x]), {x, t, 1}]NDSolve[{D[R and the error message was Author AdminPosted on February 27, 2019Categories ArticlesTags differential, DSolve, equations, NDSolve, ODEs, simultaneous Differential Equations – Complex Solutions for ODEs How do I solve the following IVP problem in Mathematica, so I get real solutions? $$Q ((t) = b – dfrac {Q (t)} {100-t}; quad Q (0) = 250$$ I tried the following: $$text { assumptions} = b> 0; text { assumptions} = t> 0;$$ $$f = text {DSolve} left[left{Q'(t)=b-frac{Q(t)}{100-t},Q(0)=250right},Q,tright][[1,1,2]]$$ $$f (t)$$ which leads to the following: $$frac {1} {2} (-2 i pi b t-2 bt log (100) -200 b log (t-100) +2 bt log (t-100) +200 i pi b + 200 b log (100) -5 t + 500)$$ Any help would be appreciated. Many Thanks!!! Author AdminPosted on January 30, 2019Categories ArticlesTags complex, differential, equations, ODEs, Solutions approx. classical analysis and odes – Sturm-Liouville equation with finite number of eigenvalues? Consider the following Storm Liouville (SL) intrinsic value problem $$x in (- infty, 0)$$ $$(py & # 39;) & # 39; – qy = – lambda ^ 2wy,$$ in which $$p (x) = x ^ 2$$, $$w (x) = 1$$, and $$q (x) = (x / 2 + a) ^ 2 + a$$ with parameters $$a> 0$$, It has a regular singularity $$x = 0$$, We basically hope for something like a homogeneous Dirichlet b.c. It is solved by the substitution $$y (x) = e ^ {x / 2} x ^ {- frac {1} {2} + sqrt {(a + frac {1} {2}) ^ 2- lambda ^ 2}} u (x)$$This leads directly to a standard confluent hypergeometric equation $$xu & # 39; & # 39; (x) + ( gamma-x) u & # 39; (x) – alphau (x) = 0,$$ in which $$alpha = sqrt {(a + frac {1} {2}) ^ 2- lambda ^ 2} -a + frac {1} {2}$$ and $$gamma = 1 + 2 sqrt {(a + frac {1} {2}) ^ 2- lambda ^ 2}$$, There are two independent solutions (1st type & 2nd type).Let's take a few ubiquitous arguments when solving the eigensystem related to confluent hypergeometric equations. Non-divergence required $$x = 0$$the 2nd kind is dropped. Non-divergence required $$infty$$the 1st kind is reduced to a polynomial if $$– alpha$$ is a non-negative integer and eigenvalue $$lambda ^ 2$$ is reached. However, seen from this condition for $$alpha$$obviously We have only a finite and finite series of eigenvalues ​​that is different from the infinite eigen spectrum that the SL theory claims. What is wrong here? Do I miss a few solutions? Author AdminPosted on January 5, 2019Categories ArticlesTags analysis, approx, classical, eigenvalues, equation, finite, number, ODEs, SturmLiouville Differential Equations – Solving a Nonlinear System of 2nd Order ODEs with NDSolve I'm trying to solve a second-order ODE system with five dependent variables. The equations are: eqn1 = D[z]* C & # 39; & # 39;[z] - u * C & # 39;[z] == a * r1[z] eqn2 = k[z]* T & # 39; & # 39;[z] - q * T & # 39;[z] == a * (r1[z]* b + r2[z]* c + r3[z]* d) from where A B C D are constants; k[z], D[z]ri[z] (from where i = 1,2,3) are dependent variables of T[z] and C[z], and C[z] and T[z] depend on each other and where r1[z] = (kc1[z]* keb[z]* (y1[z]* p - (y3[z]* p + y4[z]* p) / keq[z]* P)) / (1 + keb[z]* y1[Z*p+kh2[Z*p+kh2[z*p+kh2[z*p+kh2[z]* y4[z]* p + kst[z]* y3[z]* p) ^ 2 r2 and r3 are like r1[z], I have the following constraints: initConds = {T[0] == 893.15, T & # 39;[L] == 0, C[0] 0.992, C & # 39;[L] == 0}; from where L = 9, I use it to use NDSolve as shown below, but the code did not trigger the system, and the error message was also displayed. s = NDSolve[{}{TeqnsinitConds[{}{TeqnsinitConds[{eqnsinitConds}{T[{eqnsinitConds}{T[z]C[z]}, {z, 0, 9, 0.01}] NDSolve :: ntdv: Unable to resolve to find an explicit formula for the derivatives. Consider using the Method -> {"EquationSimplification" -> "Residual"} option. Suppose Mathematica can solve my system, how could I write code? NDSolve or another method? Author AdminPosted on December 31, 2018Categories ArticlesTags 2nd, differential, equations, NDSolve, nonlinear, ODEs, order, solving, system Posts navigation Page 1 Page 2 Next page ```