# Solving a differential equation using DSolve: no output **URL:** \ **Category:** Questions\ **Tags:** calculus, equation-solving\ **Created:** [August 9, 2021, 8:09pm UTC](https://community.wolfram.com/t/solving-a-differential-equation-using-dsolve-no-output/18658 "2021-08-09T20:09:59Z")\ **Posts on this page:** 4\ **Page:** 1
**Author:** ![lstadtmueller](https://avatars.discourse-cdn.com/v4/letter/l/85e7bf/32.png) [@lstadtmueller](https://community.wolfram.com/u/lstadtmueller)\ **Post date:** [August 9, 2021, 8:09pm UTC](https://community.wolfram.com/t/solving-a-differential-equation-using-dsolve-no-output/18658/1 "2021-08-09T20:09:59Z")
Hi everyone. I try to solve the coupled differential equations that you can see in the notebook. But Mathematica can’t find a solution and just returns the 2 differential equations, even though there exists a non-trivial solution. If W1[x] and W2[x] are linear and W1[x] = (2/R0)\*W2[x] the equations are fulfilled. So, this is not the solution that I am interested in, but it shows that something is going wrong if Mathematica can’t find it. Does anyone know what it is? Best. Luca [Open in Wolfram Cloud ↗](https://www.wolframcloud.com/obj/8d4896ad-be4a-47ce-970f-ef7cd1d66ff7) ---
**Author:** ![mariuszi28cg](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/mariuszi28cg/32/308_2.png) [@mariuszi28cg](https://community.wolfram.com/u/mariuszi28cg)\ **Post date:** [August 11, 2021, 3:22pm UTC](https://community.wolfram.com/t/solving-a-differential-equation-using-dsolve-no-output/18658/2 "2021-08-11T15:22:44Z")
**DSolve[]'s** support for solving coupled differential equations is still somewhat **limited** ,so don’t be surprised if some things **don’t work** yet. A workaround: ``` ClearAll["`*"]; Remove["`*"]; eq1 =k Sech[(Sqrt[k] x)/(2 Sqrt[Dm])]^2 W1[x] + 1/2 k Sech[(Sqrt[k] x)/(2 Sqrt[Dm])]^2 Sinh[(Sqrt[k] x)/Sqrt[ Dm]] W1[x] - (2 k Sech[(Sqrt[k] x)/(2 Sqrt[Dm])]^2 W2[x])/R0 - ( k Sech[(Sqrt[k] x)/(2 Sqrt[Dm])]^2 Sinh[(Sqrt[k] x)/Sqrt[Dm]] W2[ x])/R0 + Dm W1''[x] == 0; eq2 =(Dc k R0 W1[x])/(4 Dm) + ( Dc k R0 Tanh[(Sqrt[k] x)/(2 Sqrt[Dm])] W1[x])/(2 Dm) + ( Dc k R0 Tanh[(Sqrt[k] x)/(2 Sqrt[Dm])]^2 W1[x])/(4 Dm) - ( Dc k W2[x])/(2 Dm) - (Dc k Tanh[(Sqrt[k] x)/(2 Sqrt[Dm])] W2[x])/ Dm - (Dc k Tanh[(Sqrt[k] x)/(2 Sqrt[Dm])]^2 W2[x])/(2 Dm) + Dc W2''[x] == 0; EQ = eq1 /. Solve[eq2, W1[x]] /. D[Solve[eq2, W1[x]], {x, 2}] // FullSimplify ``` Solution for W2(x) : ``` SOL = DSolve[EQ[[1]], W2[x], x] // Simplify (*{{W2[x] -> C[3] + x C[4] + 1/6 ((3 Dm E^((Sqrt[k] x)/Sqrt[Dm]) C[1])/k - (6 Dm C[1])/( k + E^((Sqrt[k] x)/Sqrt[Dm]) k) + 6 x^2 C[1] - ( Dm C[2])/((1 + E^((Sqrt[k] x)/Sqrt[Dm]))^2 k) + (4 Dm C[2])/( k + E^((Sqrt[k] x)/Sqrt[Dm]) k) - (2 Sqrt[Dm] x C[2])/Sqrt[ k] - (6 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[Dm])]^2)/k + ( 2 Dm C[2] Log[1 + E^((Sqrt[k] x)/Sqrt[Dm])])/k + ( 6 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[ Dm])] ((3 + 4 E^((Sqrt[k] x)/Sqrt[Dm]))/(1 + E^((Sqrt[k] x)/Sqrt[ Dm]))^2 + 2 Log[1 + E^((Sqrt[k] x)/Sqrt[Dm])]))/k + ( 12 Dm C[1] PolyLog[2, -E^(((Sqrt[k] x)/Sqrt[Dm]))])/k)}}*) ``` Solution for W1(x) : ``` (Solve[eq2, W1[x]] /. SOL /. D[SOL, {x, 2}])[[1, 1, 1]] // Simplify (*{W1[x] -> 1/(12 k R0) Sech[(Sqrt[k] x)/( 2 Sqrt[Dm])]^2 (-3 Dm C[1] + 12 k x^2 C[1] + Dm C[2] - 4 Sqrt[Dm] Sqrt[k] x C[2] + 12 k C[3] + 12 k x C[4] + 3 Dm C[1] Cosh[(2 Sqrt[k] x)/Sqrt[Dm]] + 6 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[Dm])] - 12 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[Dm])]^2 + 4 Dm C[2] Log[1 + E^((Sqrt[k] x)/Sqrt[Dm])] + 24 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[Dm])] Log[ 1 + E^((Sqrt[k] x)/Sqrt[Dm])] + Cosh[(Sqrt[k] x)/Sqrt[ Dm]] (12 k x^2 C[1] + 3 Dm C[2] - 4 Sqrt[Dm] Sqrt[k] x C[2] + 12 k C[3] + 12 k x C[4] - 12 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[Dm])]^2 + 4 Dm C[2] Log[1 + E^((Sqrt[k] x)/Sqrt[Dm])] + 6 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[ Dm])] (3 + 4 Log[1 + E^((Sqrt[k] x)/Sqrt[Dm])])) + 48 Dm C[1] Cosh[(Sqrt[k] x)/(2 Sqrt[Dm])]^2 PolyLog[ 2, -E^(((Sqrt[k] x)/Sqrt[Dm]))] - 12 Dm C[1] Sinh[(Sqrt[k] x)/Sqrt[Dm]] - 3 Dm C[2] Sinh[(Sqrt[k] x)/Sqrt[Dm]] - 18 Dm C[1] Log[E^((Sqrt[k] x)/Sqrt[Dm])] Sinh[(Sqrt[k] x)/Sqrt[ Dm]])}*) ``` ---
**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\ **Post date:** [August 11, 2021, 7:14pm UTC](https://community.wolfram.com/t/solving-a-differential-equation-using-dsolve-no-output/18658/3 "2021-08-11T19:14:33Z")
An approach using a change of dependent variable: ``` origEqs = {(k + k Tanh[(Sqrt[k] x)/(2 Sqrt[Dm])] - k Tanh[(Sqrt[k] x)/(2 Sqrt[Dm])]^2) (W1[x] - (2 W2[x])/R0) + Dm (W1^\[Prime]\[Prime])[x] == 0, (Dc k R0 (1 + Tanh[(Sqrt[k] x)/(2 Sqrt[Dm])])^2 (W1[x] - ( 2 W2[x])/R0))/(4 Dm) + Dc (W2^\[Prime]\[Prime])[x] == 0}; substitution = W1 -> Function[x, (2 W2[x])/R0 + f[x]]; newEqs = Solve[origEqs /. substitution, {f''[x], W2''[x]}][[1]] /. Rule -> Equal solF = DSolve[newEqs[[1]], f, x][[1]] solW2 = DSolve[newEqs[[2]] /. solF, W2[x], x][[1]] // PowerExpand // Simplify W1[x] -> (W1[x] /. substitution /. solF) ``` It is disappointing that Mathematica cannot solve the two new equation together ``` DSolve[newEqs, {f, W2}, x] ``` so that I am forced to solve separately the first equation and then replace into the second. ---
**Author:** ![lstadtmueller](https://avatars.discourse-cdn.com/v4/letter/l/85e7bf/32.png) [@lstadtmueller](https://community.wolfram.com/u/lstadtmueller)\ **Post date:** [August 12, 2021, 3:43pm UTC](https://community.wolfram.com/t/solving-a-differential-equation-using-dsolve-no-output/18658/4 "2021-08-12T15:43:56Z")
Thank you very much for your effort. I tried to convert the problem to one that Mathematica could handle but without success. So, thanks. Having your approaches is really helpful.