ODE problem using DSolve

I would like to use DSolve (or NDSolve) to verify that the solution to the ODE problem

-4(v''[t]+(2/r)v'[t])-2*v[t]*Log[v[t]]-(3+(3/2)Log[4 Pi])*v[t]==0, 

with conditions $ \lim_{t\to \infty}v(t)=0$ and $ v'(0)=0$ is given by

v[t]=(4 Pi)^(-3/4)*Exp[-t^2/8]. 

I am able to verify this by hand, but am having trouble using Mathematica to verify it. I would like to use Mathematica to solve this differential equation, and later on modify some terms in the ODE to see how the solution changes.

Perhaps I am making a foolish mistake. I have also tried using NDSolve, but did not obtain the correct solution. I would appreciate any tips. Below you can find the picture of the error messages. Thanks for your help.

Picture of output

sol=DSolve[{-4(v''[t]+(2/r)v'[t])-2*v[t]*Log[v[t]] -(3+(3/2)Log[4 Pi])*v[t]==0,v[Infinity]==0,v'[0]==0},v[t],t] Plot[Evaluate[v[t] /. sol], {t, 0, 10}, PlotRange -> All] 

Why both DSolve and NDSolve are unable to solve a second-order differential equation?

I am trying to solve a recurrence relation using generating functions method. After some long calculations, I have arrived to this second-order differential equation: $ 0.5 x^5 y”(x)+(2x^4+x^3)y'(x)+\left(x^3+x^2+x-1\right)y(x)+1=0$

and these conditions: $ y(0)=1, y'(0)=1$ . $ y(x)$ is the function that needs to be expanded as Taylor Series at $ x=0$ to obtain the sequence from the coefficients. However, when I try to solve it both using DSolve and NDSolve, I have no luck. With DSolve it just returns the request itself:

$ $ \text{DSolve}\left[\left\{0.5 x^5 y”(x)+(2. x+1) x^3 y'(x)+\left(1. x^3+x^2+x-1\right)y(x)+1=0,y(0)=1,y'(0)=1\right\},y,x\right]$ $

And with NDSolve I just receive errors and no equation:

Power::infy: Infinite expression 1/0.^5 encountered. Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered. NDSolve::ndnum: Encountered non-numerical value for a derivative at x == 0.`. 

$ \text{NDSolve}\left[\left\{0.5 x^5 y”(x)+(2. x+1) x^3 y'(x)+\left(1. x^3+x^2+x-1\right)y(x)+1=0,y(0)=1,y'(0)=1\right\},y,\{x,0,1\}\right]$

How could I resolve this problem?

Solving Laplace PDE with DSolve

I’m trying to get an analytical solution of Laplace PDE with Dirichlet boundary conditions (in polar coordinates). I managed to solve it numerically with NDSolveValue and I know there is an analytical solution and I know what it is, but I would like DSolve to return it. But DSolve returns the input.

sol = DSolve[{Laplacian[       u[\[Rho], \[CurlyPhi]], {\[Rho], \[CurlyPhi]}, "Polar"] == 0,     DirichletCondition[u[\[Rho], \[CurlyPhi]] == 0,       1 <= \[Rho] <= 2 && \[CurlyPhi] == 0],     DirichletCondition[u[\[Rho], \[CurlyPhi]] == 0,       1 <= \[Rho] <= 2 && \[CurlyPhi] == \[Pi]],      DirichletCondition[      u[\[Rho], \[CurlyPhi]] == Sin[\[CurlyPhi]], \[Rho] == 1 &&        0 <= \[CurlyPhi] <= \[Pi]],      DirichletCondition[      u[\[Rho], \[CurlyPhi]] == 0., \[Rho] == 2 &&        0 <= \[CurlyPhi] <= \[Pi]]},     u, {\[Rho], 1, 2}, {\[CurlyPhi], 0, \[Pi]}]; 

DSolve is not working

I am trying to solve differential equation but, Mathematica is not giving any output. Any idea how to solve this or any alternate way to solve this differential equation. I appreciate your help.

DSolve[{(54 g[r] +          3 r (13 Derivative[1][f][r] + 5 Derivative[1][g][r] +             4 r (f^\[Prime]\[Prime])[r]))/        r^2 == (48 g[r])/r + (         40 (66 - 89 r + 28 r^2) Hypergeometric2F1[2 - Sqrt[3],            2 + Sqrt[3], 4, (9 r)/10])/(         40 Hypergeometric2F1[2 - Sqrt[3], 2 + Sqrt[3], 4, 9/10] +           3 Hypergeometric2F1[3 - Sqrt[3], 3 + Sqrt[3], 5, 9/10]) +          57 Derivative[1][f][r] + 13 Derivative[1][g][r] +          11 r (f^\[Prime]\[Prime])[r], (1/(        r^4))(-132 g[r] +           114 r g[r] - (r^3 (40 (-89 + 56 r) Hypergeometric2F1[2 - Sqrt[3],                  2 + Sqrt[3], 4, (9 r)/10] +                9 (66 - 89 r + 28 r^2) Hypergeometric2F1[3 - Sqrt[3],                  3 + Sqrt[3], 5, (9 r)/10]))/(40 Hypergeometric2F1[               2 - Sqrt[3], 2 + Sqrt[3], 4, 9/10] +              3 Hypergeometric2F1[3 - Sqrt[3], 3 + Sqrt[3], 5, 9/10]) -           15 r Derivative[1][f][r] + 11 r^2 Derivative[1][f][r] +           39 r Derivative[1][g][r] - 59 r^2 Derivative[1][g][r] +           15 r^2 (f^\[Prime]\[Prime])[r] -           13 r^3 (f^\[Prime]\[Prime])[r] +           27 r^2 (g^\[Prime]\[Prime])[r] -           24 r^3 (g^\[Prime]\[Prime])[r]) == 0}, {f[r], g[r]}, r] 

How to Solve $y'(x)=x\ln(y(x))$ with $y(1)=1$ using DSolve

The solution to $ y’=x \ln(y)$ with initial conditions $ y(1)=1$ is $ y=1$ .

How to persuade DSolve to obtain this solution?

ClearAll[y, x]; ode = y'[x] == x Log[y[x]]; ic = y[1] == 1; sol = DSolve[{ode, ic}, y[x], x] 

Mathematica graphics

One can see that $ y=1$ is solution that also satisfies the ic by looking at direction field.

ClearAll[x, y]; fTerm = x Log[y]; StreamPlot[ {1, fTerm}, {x, -1, 3}, {y, 0, 2},  Axes -> True,  Frame -> False,  PlotTheme -> "Classic",  AspectRatio -> 1 / GoldenRatio,  StreamPoints -> {{{{1, 1}, Red}, Automatic}},  Epilog -> {{Red, PointSize[.025], Point[{1, 1}]}},  PlotLabel -> Style[Text[Row     [{"Solution curve with initial conditions at {", 1, ",", 1,"}"}]], 14]  ] 

enter image description here

V 12.0 on windows 10

Plot the solution from DSolve

I’m trying to solve a differential equation as in the following code:

FullSimplify[DSolve[x'[t] == a + b E^(g t) + (c + d E^(-g t)) x[t], x[t], t]] 

which generates

enter image description here

Now, I would like to plot it with specific parameter values assigned, for example: a = 1; b = 2; c = 3; d = 4; g = 0.1; A = 1 where I replaced the integration constant c_1 with A.

Here is my code for plotting x[t]:

a = 1; b = 2; c = 3; d = 4; g = 0.1; A = 1 x[t_] := E^(-((d E^(-g t))/g) + c t) (A + Integrate[E^((d E^(-g K[1]))/g - c K[1]) (a + b E^(g K[1])), {K[1], 1, t}]) Plot[x[t], {t, 1, 10}] 

It runs forever. To check whether Mathematica is doing calculations, I tried

x[1] 

and it yielded

3.83926*10^-15 

which is nice. But when I tried

x[2] 

I got

enter image description here

It seems Mathematica cannot compute the integral unless the integration region is $ \int_1^1$ . Is this because the integrand is too complicated? Is there any way to let Mathematica compute it? Thanks!

Workaround for DSolve in V 12 when it gives undefined as solution to 1D heat PDE?

Comparing the following, all done from clean kernel


Mathematica graphics


Mathematica graphics

The strange thing is that V 12 can solve this same PDE without the assumptions

k    = 1/10; A = 60; pde  = D[u[x, t], t] == k*D[u[x, t], {x, 2}]; bc   = u[0, t] == A; ic   = u[x, 0] == 0; sol  = DSolve[{pde, bc, ic}, u[x, t], {x, t}] 

Mathematica graphics

But it says in the above answer that it wants x>0,t>0, which is why I gave it the assumptions to help it, but then it returns undefined.

Something seems to have gone wrong in V 12 DSolve here, or may be in the Integrate? I do not know.

Do others see the same result on V 12?. Answer given by V 11.3 is the correct one.

Any workaround for V 12 to make it give same answer as V 11.3?