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DSolve not giving me an answer

Henric Larsson
Posted 1 year ago

I'm confused why DSolve can't solve this problem:

DSolve[{y'[x] == (7 x + 4 y[x])/(7 x + 4 y[x] + 4), y[-1] == -1}, 
 y[x], x]

Can you let me know how I can solve this using Mathematica?
POSTED BY: Henric Larsson
7 Replies
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POSTED BY: Michael Rogers

Thanks to all for your help. The solution can be found by using substitution u = 7x + 4y. Not sure if the algorithm can be improved to handle these problems.
POSTED BY: Henric Larsson
Updating Name
Updating Name
Posted 1 year ago

DSolve solves your ode

DSolve[{y'[x] == (7 x + 4 y[x])/(7 x + 4 y[x] + 4)}, y[x], x]

as an Abel equation of the second kind. The problem is with solving the initial condition for the parameter of integration. Not sure how the substitution u = 7 x + 4y helps with that.
POSTED BY: Updating Name

With substitution u = 7 x + 4y not helps.

POSTED BY: Mariusz Iwaniuk
POSTED BY: Michael Rogers

The numerical solution with NDSolve misses a singular point with infinite derivative. Here is a derivation of the solution:

diffEq = y'[x] == (7  x + 4  y[x])/(7  x + 4  y[x] + 4);
solY = DSolveValue[diffEq, y, x]
eq = y == solY[x]
extractProductLog = Solve[eq, Cases[eq, _ProductLog, All][[1]]][[1, 1]]
implicitEq = 
 extractProductLog /. (ProductLog[z_] -> w_) :> z == w  Exp[w]
initialCond = implicitEq /. {x -> -1, y -> -1}
solForC1 = Solve[initialCond, C[1], Reals]
implicitEquationForInitialConditions = 
 Simplify[implicitEq /. solForC1[[1]]]
Reduce[implicitEquationForInitialConditions, y, Reals]
ContourPlot[implicitEquationForInitialConditions,
 {x, -3, 3}, {y, -3, 3}]
POSTED BY: Gianluca Gorni

Looks like to me is a bug in DSolve. Solution is:

$y(x)=-\frac{7}{11}-\frac{7 x}{4}+\frac{4}{11} W_{-1}\left(\frac{1}{16} (-93) e^{\frac{7}{4}+\frac{121 x}{16}}\right)$

eq = {y'[x] == (7 x + 4 y[x])/(4 + 7 x + 4 y[x]), y[-1] == -1};
eq /. y -> 
   Function[{x}, -(7/11) - (7 x)/4 + 
     4/11  ProductLog[-1, -(93/16)  E^(
        7/4 + (121 x)/16)]] // FullSimplify

(*{True, True}*)

Regards M.I
POSTED BY: Mariusz Iwaniuk
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