# Why can't DSolve find a solution for my system of ODEs?

I want to find a basis for the set of solutions of the following system of ODEs.

DSolve[{x'[t] == x[t]/t - y[t], y'[t] == x[t] + y[t]/t}, {y[t], x[t]}, t]


I get very unexpected result when I evaluate the above expression. Mathematica returns my input.

Of course, I know the answer.

Update

From Wolfram Technical Support:

Thank you for contacting Wolfram Technical Support. I have confirmed that DSolve is not able to find the solution to the equation you provided. I found a workaround by introducing an arbitrary variable 'a' then setting a=-1 at the end of the calculation. This can be done as follows:

X[t_]:={x[t],y[t]} A={{1/t,a},{-a,1/t}}; system=X'[t]==A.X[t]; sol=DSolve[system,{x,y},t] sol/.{a-> -1}

DSolve should be able to sole the equation in the original form you provided. I will file a report with our developers regarding this issue. Thank you for bringing this to our attention. [CASE:3808703]

• Does the answer that you know work when t==0? When it can be justified, eliminating denominators seems to almost always give better results. – Bill Jan 1 '17 at 3:44

Fixed in version 11.1 Now DSolve can solve it directly
 $Version  11.1.0 for Microsoft Windows (64-bit) (March 13, 2017)  ClearAll[x,t,y]; DSolve[{x'[t]==x[t]/t-y[t],y'[t]==x[t]+y[t]/t},{y[t],x[t]},t] {{x[t]->t C Cos[t]-t C Sin[t],y[t]->t C Cos[t]+t C Sin[t]}}  I'm not sure why this doesn't work. Until someone finds the reason, here is a workaround to solve the system of equations: ode={x'[t] == x[t]/t - y[t], y'[t] == x[t] + y[t]/t}  Using the change of variables$x(t)=t X(t)$and$y(t)=tY(t), ode /. {x -> ((# X[#]) &), y -> ((# Y[#]) &)} (* X[t] + t X'[t] == X[t] - t Y[t], Y[t] + t Y'[t] == t X[t] + Y[t] *)  which is now easily solved by DSolve: DSolve[%, {X[t], Y[t]}, {t}] (* X[t] -> C Cos[t] - C Sin[t], Y[t] -> C Cos[t] + C Sin[t] *)  • I would read this post as a proof that there is nothing intrinsically wrong with the system of ODE's; the problem seems to be with MM, not with the equations themselves. I would be very happy if someone finds the reason for MM to dislike the equations in their initial form. – AccidentalFourierTransform Jan 1 '17 at 12:53 Here is another way to solve in Mathematica. \begin{align*} x^{\prime}\left( t\right) & =\frac{x\left( t\right) }{t}-y\left( t\right) \\ y^{\prime}\left( t\right) & =x\left( t\right) -\frac{y\left( t\right) }{t} \end{align*} Write as $$\dot{X}=AX$$ WhereA= \begin{pmatrix} \frac{1}{t} & -1\\ 1 & \frac{-1}{t} \end{pmatrix} $. Let us assume the solution is$X\left( t\right) =X\left( 0\right) e^{\int_{0}^{t}A\left( \tau\right) d\tau}$. However, this only works (since this is matrix time varying system) if$A$commutes with$e^{\int_{0}^{t}A\left( \tau\right) d\tau}$. i.e. if $$A\left( t\right) e^{\int_{0}^{t}A\left( \tau\right) d\tau}=e^{\int_{0} ^{t}A\left( \tau\right) d\tau}A\left( t\right)$$ If it does not commute, things get little more complicated. We can check, using Mathematica, that it does indeed commute in this example, so we are lucky (mat = {{1/t, -1}, {1, 1/t}}) // MatrixForm (bmat = MatrixExp[Integrate[mat, t]]) // MatrixForm mat.bmat == bmat.mat Therefore, we can ask Mathematica now to solve it, using MatrixExp x0 = {c1, c2}; (*constant of integration*) (sol = x0.bmat) // MatrixForm The above is the solution. First row is$x(t)$and second row is$y(t)\$.