I was trying to verify my manual solution with Mathematica, and it gives me an additional solution, which I don't understand how it came about. Nor can I verify the second solution.

Here is the pde specification. Solve $ w (x, t) $

$$

frac { partial w} { partial t} +3 t frac { partial w} { partial x} = w (x, t) tag {1}

$$

with initial conditions $ w (x, 0) = f (x) $.

Here is the code (Using `w`

in DSolve instead of standard `w(x,t)`

to get the solution using `Function`

to make it easier to verify)

```
ClearAll("Global`*");
pde = D(w(x, t), t) + 3*t*D(w(x, t), x) == w(x, t);
ic = w(x, 0) == f(x);
sol = DSolve({pde, ic}, w, {x, t}) (*Assumptions -> t > 0 has no effect*)
```

Gives

It is the first solution above which I have doubts that it is correct. Note that add `Assumptions -> t > 0`

to `DSolve`

It has no effect The same solutions are obtained.

Now, the second solution verifies OK, but not the first.

```
Assuming(t > 0, Simplify(pde /. sol((1, 1))))
```

```
Assuming(t > 0, Simplify(pde /. sol((2, 1))))
```

The question is: Is the first solution above correct? If so, why isn't it verified and how did Mathematica get it? Below is the manual solution.

## Appendix

My manual solution was trying to verify using Mathematica:

Leave $ w equiv w left (x left (t right), t right) $ so

begin {equation}

frac {dw} {dt} = frac { partial w} { partial x} frac {dx} {dt} + frac { partial

w} { partial t} tag {2}

end {equation}

Compare (1,2) shows that

begin {align}

frac {dw} {dt} & = w tag {3} \

frac {dx} {dt} & = 3t tag {4}

end {align}

Solve (3) gives

begin {equation}

w = Ce ^ t nonumber

end {equation}

From initial conditions in $ t = 0 $, the above becomes $ f left (x left (

0 right) right) = C $. Therefore, the above becomes

begin {equation}

w left (x, t right) = f left (x left (0 right) right) e ^ {t} tag {5}

end {equation}

From (4)

begin {align *}

x & = frac {3} {2} t ^ {2} + x left (0 right) \

x left (0 right) & = x- frac {3} {2} t ^ {2}

end {align *}

Substituting the above in (5) da

$$

w left (x left (t right), t right) = f left (x- frac {3} {2} t ^ {2} right)

e t

$$