<Text-field style="Heading 1" layout="Heading 1">Linear equations in total derivatives</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Example: Symmetries of KdV</Text-field>

coordinates([t,x], [u], 5);

NjZJJXVfNXRHNiJJJ3VfNHRfeEdGJEkodV90dHR4eEdGJEkodV90dHh4eEdGJEkndV90XzR4R0YkSSV1XzV4R0YkSSV1XzR0R0YkSSd1X3R0dHhHRiRJJ3VfdHR4eEdGJEkndV90eHh4R0YkSSV1XzR4R0YkSSZ1X3R0dEdGJEkmdV90dHhHRiRJJnVfdHh4R0YkSSZ1X3h4eEdGJEkldV90dEdGJEkldV90eEdGJEkldV94eEdGJEkkdV90R0YkSSR1X3hHRiQ=

equation (u_t = u_xxx + 6*u*u_x);

L0kkdV90RzYiLCZJJnVfeHh4R0YkIiIiKiZJInVHRiRGJ0kkdV94R0YkRiciIic=

Symmetries (more precisely, their generating functions U ) are solutions of the determining equation S = 0 where S is the universal linearization operator:

S := symmetries(u = U);

LCotSSNUREc2IjYkSSJVR0YlSSJ0R0YlIiIiKiZGJ0YpSSR1X3hHRiVGKSEiJyomLUYkNiRGJ0kieEdGJUYpSSJ1R0YlRilGLC1GJDYkRicqJEYwIiIkISIi

The determining equation for generating functions of conservation laws is L = 0, where

L := laws(1 = V);

LCgtSSNUREc2IjYkSSJWR0YlSSJ0R0YlISIiKiZJInVHRiUiIiItRiQ2JEYnSSJ4R0YlRiwiIictRiQ2JEYnKiRGLyIiJEYs

As a smooth function, U can depend only on a \357\254\201nite number of variables, i.e., there must be an upper bound on the jet order of U.

Assume that the bound is \357\254\201ve. A \357\254\201fth-order generating function is one that depends on x, t, u, ux , . . . , u5x .

Accordingly, our assumption is made known to the computer as follows:

dependence(U(x,t,u,u_x,u_xx,u_xxx,u_4x,u_5x));

L0kiVUc2IjwqSSJ0R0YkSSJ1R0YkSSJ4R0YkSSR1X3hHRiRJJXVfeHhHRiRJJnVfeHh4R0YkSSV1XzR4R0YkSSV1XzV4R0Yk

To run in automatic regime, Jets also needs to know what are the unknowns:

unknowns(U);

SSJVRzYi

Now launch the computation:

run(S);

<0> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJEkldV81eEdGJSIiIyIiIQ==

<0> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkldV80eEdGJSIiIkkldV81eEdGJUYqIiIh

<0> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkmdV94eHhHRiUiIiJJJXVfNXhHRiVGKiIiIQ==

<0> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkldV94eEdGJSIiIkkldV81eEdGJUYqIiIh

<1> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkkdV94R0YlIiIiSSV1XzV4R0YlRioiIiE=

<1> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkidUdGJSIiIkkldV81eEdGJUYqIiIh

<1> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkieEdGJSIiIkkldV81eEdGJUYqIiIh

<2> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJEkldV80eEdGJSIiIyIiIQ==

<2> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkldV80eEdGJSIiIkkmdV94eHhHRiVGKiIiIQ==

<2> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkldV94eEdGJSIiIkkldV80eEdGJUYqIiIh

<2> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkkdV94R0YlIiIiSSV1XzR4R0YlRioiIiE=

<3> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkidUdGJSIiIkkldV80eEdGJUYqIiIh

<3> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkldV80eEdGJSIiIkkieEdGJUYqIiIh

<3> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJEkmdV94eHhHRiUiIiMiIiE=

<4> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkldV94eEdGJSIiIkkmdV94eHhHRiVGKiIiIQ==

<4> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkkdV94R0YlIiIiSSZ1X3h4eEdGJUYqIiIh

<4> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkmdV94eHhHRiUiIiJJInVHRiUiIiMiIiE=

<4> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkidUdGJSIiIkkmdV94eHhHRiVGKiwkLUYkNiRGJ0kldV81eEdGJSIjNQ==

<5> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkidEdGJSIiIkkldV81eEdGJUYqLCQtRiQ2JEYnKiZJJnVfeHh4R0YlRipJInhHRiVGKiIiJA==

<5> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJEkldV94eEdGJSIiIyIiIQ==

Ly1JI3BkRzYiNiRJIlVHRiUqJkkmdV94eHhHRiUiIiJJInhHRiUiIiMiIiE=

<6> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkldV94eEdGJSIiIkkkdV94R0YlIiIjIiIh

<6> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqKEkldV94eEdGJSIiIkkidUdGJUYqSSR1X3hHRiVGKiIiIQ==

<7> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkkdV94R0YlIiIiSSV1X3h4R0YlRiosJC1GJDYkRidJJXVfNXhHRiUiIz8=

<7> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkldV94eEdGJSIiIkkidUdGJSIiIyIiIQ==

<7> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkidUdGJSIiIkkldV94eEdGJUYqLCQtRiQ2JEYnSSV1XzR4R0YlIiIp

<8> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkidEdGJSIiIkkldV80eEdGJUYqLCQtRiQ2JEYnKiZJJXVfeHhHRiVGKkkieEdGJUYqIiIk

<8> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJEkkdV94R0YlIiIkIiIh

Ly1JI3BkRzYiNiRJIlVHRiUqJkkldV94eEdGJSIiIkkieEdGJSIiIyIiIQ==

<9> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkkdV94R0YlIiIjSSJ1R0YlIiIiIiIh

<9> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJEkkdV94R0YlIiIjLCQtRiQ2JEYnSSV1XzR4R0YlIiM3

<10> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkidUdGJSIiIkkkdV94R0YlRiosJC1GJDYkRidJJnVfeHh4R0YlIiIn

<10> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkmdV94eHhHRiUiIiJJInhHRiVGKiIiIQ==

<11> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkidEdGJSIiIkkmdV94eHhHRiVGKiwkLUYkNiRGJyomSSR1X3hHRiVGKkkieEdGJUYqIiIk

<11> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJEkidUdGJSIiJSIiIQ==

Ly1JI3BkRzYiNiRJIlVHRiUqJkkieEdGJSIiI0kkdV94R0YlIiIiIiIh

<12> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJEkidUdGJSIiJCwkLUYkNiRGJ0kldV80eEdGJSIjSw==

<12> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJEkidUdGJSIiIywmKiYtRiQ2JEYnSSV1XzV4R0YlIiIiSSR1X3hHRiVGMCEjPy1GJDYkRidJJXVfeHhHRiUiIiU=

<13> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkldV94eEdGJSIiIkkieEdGJUYqIiIh

<13> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkidEdGJSIiIkkldV94eEdGJUYqLCQtRiQ2JEYnKiZJInVHRiVGKkkieEdGJUYqIiIk

<14> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkieEdGJSIiI0kidUdGJSIiIiIiIQ==

Ly1JI3BkRzYiNiRJIlVHRiVJJXVfNHhHRiUiIiE=

<14> Put:

Ly1JI3BkRzYiNiRJIlVHRiVJJXVfeHhHRiUsJComLUYkNiRGJ0kldV81eEdGJSIiIkkkdV94R0YlRi4iIz8=

<15> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkkdV94R0YlIiIiSSJ4R0YlRiosKComLUYkNiRGJ0kmdV94eHhHRiVGKkYpRiohIiQqJi1GJDYkRidJJXVfNXhHRiVGKkYwRiohIiYtRiQ2JEYnSSJ1R0YlI0YqIiIj

<15> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkidEdGJSIiIkkkdV94R0YlRiosMEYnIiInKiYtRiQ2JEYnRitGKkYrRiohIicqKEkidUdGJUYqLUYkNiRGJ0kldV81eEdGJUYqSSZ1X3h4eEdGJUYqISNJKiZGNkYqRjRGKkYxKiYtRiQ2JEYnRjdGKkY3RipGMSooRjtGKkYzRipGK0YqISM9KiZGM0YqLUYkNiRGJ0YzRioiIiQ=

<16> Put:

Ly1JI3BkRzYiNiRJIlVHRiUqJkkidUdGJSIiIkkidEdGJUYqLCotRiQ2JEYnSSJ4R0YlIiM3KiZJJHVfeEdGJSIiIy1GJDYkRidJJnVfeHh4R0YlRiohIz0qKC1GJDYkRidJJXVfNXhHRiVGKkY2RipGMkYqISNJKiZGMkYqLUYkNiRGJ0YpRioiIiQ=

Ly1JI3BkRzYiNiRJIlVHRiUqJkkieEdGJSIiIkkidUdGJUYqIiIh

<16> Put:

Ly1JI3BkRzYiNiRJIlVHRiVJInhHRiUsKComSSR1X3hHRiUiIiItRiQ2JEYnSSJ1R0YlRiwjRiwiIiMqJkYrRjEtRiQ2JEYnSSZ1X3h4eEdGJUYsISIkKigtRiQ2JEYnSSV1XzV4R0YlRixGNUYsRitGLCEiJg==

Ly1JI3BkRzYiNiRJIlVHRiUqJEkieEdGJSIiIyIiIQ==

<16> Put:

Ly1JI3BkRzYiNiRJIlVHRiVJInRHRiVRXHY2KlUqdV94LTYqdV94XjIqcGQoVSx1X3gpLTYqcGQoVSx1XzV4KSp1XzV4KnVfeC0zMCpwZChVLHVfNXgpKnVfeCp1KnVfeHh4LTE1KnBkKFUsdV81eCkqdV94eHheMi0xNSp1X3gqdV94eHgqcGQoVSx1X3h4eCktMTgqcGQoVSx1X3h4eCkqdSp1X3heMiszKnBkKFUsdSkqdSp1X3grMy8yKnVfeHh4KnBkKFUsdSlGJQ==

<17> Success!

To retrieve the results, on should proceed as follows.

dependence();

L0kiVUc2IjwpSSJ0R0YkSSJ1R0YkSSJ4R0YkSSR1X3hHRiRJJXVfeHhHRiRJJnVfeHh4R0YkSSV1XzV4R0Yk

Z := clear(pds);

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

The above command also removes all assignments to partial derivatives (otherwise Z would be a set of zeroes). This is why clear(pds) should not be called repeatedly.

<Text-field style="Heading 3" layout="Heading 3">First step</Text-field>

U := U1*u_5x + U0; dependence(U1(t,x,u,u_x,u_xx,u_xxx), U0(t,x,u,u_x,u_xx,u_xxx));

LCYqJkkjVTFHNiIiIiJJJXVfNXhHRiVGJkYmSSNVMEdGJUYm

NiQvSSNVMEc2IjwoSSJ0R0YlSSJ1R0YlSSJ4R0YlSSR1X3hHRiVJJXVfeHhHRiVJJnVfeHh4R0YlL0kjVTFHRiVGJg==

map(expand, Z);

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

map(expand@pd, Z, u_5x);

Ny8tSSNwZEc2IjYkSSNVMUdGJSokSSR1X3hHRiUiIiMiIiEtRiQ2JEYnKiRJJnVfeHh4R0YlRipGK0YrLUYkNiRGJyomRikiIiJGL0YzRissJi1GJDYkRicqJkkidUdGJUYzRilGM0YzLUYkNiRGJ0YvISInLUYkNiRGJyomRjhGM0YvRjMtRiQ2JEYnKiRGOEYqLUYkNiRGJ0kldV94eEdGJSwoLUYkNiRGJ0kieEdGJUYzKiZGKUYzLUYkNiRGJ0Y4RjMjISIiRioqJkYpRipGOUYzIiIkLC4tRiQ2JEYnSSJ0R0YlRjMqJkYpRiotRiQ2JEYnRilGMyIiJyooRjlGM0YpRjNGL0YzIiM6KihGOUYzRjhGM0YpRioiIz0qKEZKRjNGOEYzRilGMyEiJComRkpGM0YvRjMjRmduRio=

unknowns(U1,U0):

run(Z);

<0> Put:

Ly1JI3BkRzYiNiRJI1UxR0YlSSJ0R0YlIiIh

Ly1JI3BkRzYiNiRJI1UxR0YlSSJ1R0YlIiIh

Ly1JI3BkRzYiNiRJI1UxR0YlSSJ4R0YlIiIh

Ly1JI3BkRzYiNiRJI1UxR0YlKiRJInVHRiUiIiMiIiE=

Ly1JI3BkRzYiNiRJI1UxR0YlKiRJJHVfeEdGJSIiIyIiIQ==

Ly1JI3BkRzYiNiRJI1UxR0YlKiRJJnVfeHh4R0YlIiIjIiIh

Ly1JI3BkRzYiNiRJI1UxR0YlKiZJInVHRiUiIiJJJHVfeEdGJUYqIiIh

Ly1JI3BkRzYiNiRJI1UxR0YlKiZJInVHRiUiIiJJJnVfeHh4R0YlRioiIiE=

Ly1JI3BkRzYiNiRJI1UxR0YlKiZJJHVfeEdGJSIiIkkmdV94eHhHRiVGKiIiIQ==

Ly1JI3BkRzYiNiRJI1UxR0YlSSR1X3hHRiUiIiE=

Ly1JI3BkRzYiNiRJI1UxR0YlSSV1X3h4R0YlIiIh

Ly1JI3BkRzYiNiRJI1UxR0YlSSZ1X3h4eEdGJSIiIQ==

<0> Put:

Ly1JI3BkRzYiNiRJI1UwR0YlSSJ0R0YlUVx0Nip1X3gqVTArMyp1KnVfeCpwZChVMCx1KSszLzIqdV94eHgqcGQoVTAsdSktMTUqVTEqdV94eHheMi0zMCpVMSp1X3h4eCp1KnVfeC0xNSp1X3gqdV94eHgqcGQoVTAsdV94eHgpLTE4KnUqdV94XjIqcGQoVTAsdV94eHgpLTYqdV94XjIqcGQoVTAsdV94KUYl

Ly1JI3BkRzYiNiRJI1UwR0YlSSJ4R0YlLCgqJkkkdV94R0YlIiIiLUYkNiRGJ0kidUdGJUYsI0YsIiIjKiZGK0YxLUYkNiRGJ0kmdV94eHhHRiVGLCEiJCooSSNVMUdGJUYsRjVGLEYrRiwhIiY=

Ly1JI3BkRzYiNiRJI1UwR0YlKiRJJHVfeEdGJSIiIyIiIQ==

Ly1JI3BkRzYiNiRJI1UwR0YlKiRJJnVfeHh4R0YlIiIjIiIh

Ly1JI3BkRzYiNiRJI1UwR0YlKiZJInVHRiUiIiJJJHVfeEdGJUYqLCQtRiQ2JEYnSSZ1X3h4eEdGJSIiJw==

Ly1JI3BkRzYiNiRJI1UwR0YlKiZJJHVfeEdGJSIiIkkmdV94eHhHRiVGKiIiIQ==

Ly1JI3BkRzYiNiRJI1UwR0YlKiZJJXVfeHhHRiUiIiJJJnVfeHh4R0YlRioiIiE=

Ly1JI3BkRzYiNiRJI1UwR0YlKiZJJnVfeHh4R0YlIiIiSSJ1R0YlIiIjIiIh

Ly1JI3BkRzYiNiRJI1UwR0YlKiZJJnVfeHh4R0YlIiIjSSJ1R0YlIiIiIiIh

<0> Put:

Ly1JI3BkRzYiNiRJI1UwR0YlKiRJInVHRiUiIiMsJComSSNVMUdGJSIiIkkkdV94R0YlRi4iI2c=

Ly1JI3BkRzYiNiRJI1UwR0YlKiRJInVHRiUiIiQiIiE=

Ly1JI3BkRzYiNiRJI1UwR0YlKiRJJXVfeHhHRiUiIiMiIiE=

Ly1JI3BkRzYiNiRJI1UwR0YlKiZJInVHRiUiIiJJJXVfeHhHRiVGKiIiIQ==

Ly1JI3BkRzYiNiRJI1UwR0YlKiZJInVHRiUiIiJJJnVfeHh4R0YlRiosJEkjVTFHRiUiIzU=

Ly1JI3BkRzYiNiRJI1UwR0YlKiZJJXVfeHhHRiUiIiJJInVHRiUiIiMiIiE=

<0> Put:

Ly1JI3BkRzYiNiRJI1UwR0YlSSV1X3h4R0YlLCQqJkkjVTFHRiUiIiJJJHVfeEdGJUYsIiM/

<1> Success!

dependence();

NiQvSSNVMEc2IjwoSSJ0R0YlSSJ1R0YlSSJ4R0YlSSR1X3hHRiVJJXVfeHhHRiVJJnVfeHh4R0YlL0kjVTFHRiU8Ig==

Z := clear(pds);

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

<Text-field style="Heading 3" italic="false" layout="Heading 3">Second step</Text-field>

unksselect(Z, {U0});

NyYtSSNwZEc2IjYkSSNVMEdGJSokSSR1X3hHRiUiIiMtRiQ2JEYnKiRJJnVfeHh4R0YlRiotRiQ2JEYnKiZGKSIiIkYuRjIsJi1GJDYkRicqJkkidUdGJUYyRilGMkYyLUYkNiRGJ0YuISIn

U0 := U03*u_x+U04*u_xxx+U05; dependence(U03(t, u, x, u_xx), U04(t, u, x, u_xx), U05(t, u, x, u_xx));

LCgqJkkkVTAzRzYiIiIiSSR1X3hHRiVGJkYmKiZJJFUwNEdGJUYmSSZ1X3h4eEdGJUYmRiZJJFUwNUdGJUYm

NiYvSSRVMDNHNiI8JkkidEdGJUkidUdGJUkieEdGJUkldV94eEdGJS9JJFUwNEdGJUYmL0kkVTA1R0YlRiYvSSNVMUdGJTwi

map(simpl, {Diff(U0, u_x, u_x), Diff(U0, u_xxx, u_x), Diff(U0, u_xxx, u_xxx)});

PCMiIiE=

map(simpl,Z);

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

unknowns(U1,U03, U04, U05):

run(Z);

<0> Put:

L0kkVTA0RzYiLCQtSSNwZEdGJDYkSSRVMDNHRiRJInVHRiQjIiIiIiIn

Ly1JI3BkRzYiNiRJJFUwNEdGJSokSSJ1R0YlIiIjIiIh

Ly1JI3BkRzYiNiRJJFUwNEdGJSomSSJ1R0YlIiIiSSV1X3h4R0YlRioiIiE=

Ly1JI3BkRzYiNiRJJFUwNEdGJSomSSV1X3h4R0YlIiIiSSJ4R0YlRioiIiE=

Ly1JI3BkRzYiNiRJJFUwNEdGJUkldV94eEdGJSIiIQ==

<0> Put:

Ly1JI3BkRzYiNiRJJFUwM0dGJSokSSJ1R0YlIiIkIiIh

Ly1JI3BkRzYiNiRJJFUwM0dGJSomSSJ1R0YlIiIiSSV1X3h4R0YlRioiIiE=

Ly1JI3BkRzYiNiRJJFUwM0dGJSomSSV1X3h4R0YlIiIiSSJ1R0YlIiIjIiIh

Ly1JI3BkRzYiNiRJJFUwM0dGJSooSSJ4R0YlIiIiSSJ1R0YlRipJJXVfeHhHRiVGKiIiIQ==

<0> Put:

Ly1JI3BkRzYiNiRJJFUwM0dGJSokSSJ1R0YlIiIjLCRJI1UxR0YlIiNn

Ly1JI3BkRzYiNiRJJFUwNUdGJSokSSV1X3h4R0YlIiIjLCQqJi1GJDYkSSRVMDNHRiVGKCIiIkkkdV94R0YlRjAhIiI=

Ly1JI3BkRzYiNiRJJFUwNUdGJSomSSV1X3h4R0YlIiIiSSJ1R0YlIiIjIiIh

<0> Put:

Ly1JI3BkRzYiNiRJJFUwM0dGJSokSSV1X3h4R0YlIiIjIiIh

Ly1JI3BkRzYiNiRJJFUwM0dGJSomSSJ4R0YlIiIiSSJ1R0YlRioiIiE=

Ly1JI3BkRzYiNiRJJFUwM0dGJUkldV94eEdGJSwkSSNVMUdGJSIjPw==

Ly1JI3BkRzYiNiRJJFUwNUdGJSokSSJ1R0YlIiIjIiIh

Ly1JI3BkRzYiNiRJJFUwNUdGJSomSSJ1R0YlIiIiSSV1X3h4R0YlRioiIiE=

<0> Put:

Ly1JI3BkRzYiNiRJJFUwNUdGJUkidUdGJSwkLUYkNiRJJFUwM0dGJSomRigiIiJJInRHRiVGLiNGLiIiKg==

Ly1JI3BkRzYiNiRJJFUwNUdGJSomSSV1X3h4R0YlIiIiSSJ4R0YlRioiIiE=

Ly1JI3BkRzYiNiRJJFUwNUdGJUkldV94eEdGJSIiIQ==

<0> Put:

L0kkVTA1RzYiLCYtSSNwZEdGJDYkSSRVMDNHRiRJInRHRiQjIiIiIiInKiZJInVHRiRGLC1GJzYkRilJInhHRiRGLCEiIg==

Ly1JI3BkRzYiNiRJJFUwM0dGJSomSSJ1R0YlIiIiSSJ0R0YlRiosJC1GJDYkRidJInhHRiUiIz0=

<0> Put:

Ly1JI3BkRzYiNiRJJFUwM0dGJSokSSJ4R0YlIiIjIiIh

Ly1JI3BkRzYiNiRJJFUwM0dGJSomSSJ0R0YlIiIiSSJ4R0YlRioiIiE=

<0> Put:

Ly1JI3BkRzYiNiRJJFUwM0dGJSokSSJ0R0YlIiIjIiIh

<0> Success!

dependence();

NiQvSSRVMDNHNiI8JkkidEdGJUkidUdGJUkieEdGJUkldV94eEdGJS9JI1UxR0YlPCI=

<Text-field style="Heading 3" layout="Heading 3">Third step</Text-field>

Z := clear(pds);

NyktSSNwZEc2IjYkSSRVMDNHRiUqJEkieEdGJSIiIy1GJDYkRicqJEkidEdGJUYqLUYkNiRGJyomRi4iIiJGKUYyLUYkNiRGJyomRilGMkkidUdGJUYyLCZJI1UxR0YlISNnLUYkNiRGJyokRjZGKkYyLCYtRiQ2JEYnSSV1X3h4R0YlRjJGOCEjPywmLUYkNiRGJyomRjZGMkYuRjJGMi1GJDYkRidGKSEjPQ==

unksselect(Z, {U03});

NyctSSNwZEc2IjYkSSRVMDNHRiUqJEkieEdGJSIiIy1GJDYkRicqJEkidEdGJUYqLUYkNiRGJyomRi4iIiJGKUYyLUYkNiRGJyomRilGMkkidUdGJUYyLCYtRiQ2JEYnKiZGNkYyRi5GMkYyLUYkNiRGJ0YpISM9

lprint(map(convert, {Diff(U03, t, t), Diff(U03, x, t), Diff(U03, x, u), Diff(U03, x, x)}, diff));

{diff(U03(t, u, x, u_xx), t, t), diff(U03(t, u, x, u_xx), t, x), diff(U03(t, u, x, u_xx), x, u), diff(U03(t, u, x, u_xx), x, x)}

U03 := U031*t+U032*x+U033; dependence(U031(u, u_xx),U032(u_xx),U033(u, u_xx));

LCgqJkklVTAzMUc2IiIiIkkidEdGJUYmRiYqJkklVTAzMkdGJUYmSSJ4R0YlRiZGJkklVTAzM0dGJUYm

NiYvSSNVMUc2IjwiL0klVTAzMUdGJTwkSSJ1R0YlSSV1X3h4R0YlL0klVTAzMkdGJTwjRisvSSVVMDMzR0YlRik=

map(simpl, {Diff(U03, t, t), Diff(U03, x, t), Diff(U03, x, u), Diff(U03, x, x)});

PCMiIiE=

map(simpl, Z);

NykiIiFGI0YjRiMsKEkjVTFHNiIhI2cqJi1JI3BkR0YmNiRJJVUwMzFHRiYqJEkidUdGJiIiIyIiIkkidEdGJkYwRjAtRio2JEklVTAzM0dGJkYtRjAsKiomLUYqNiRGLEkldV94eEdGJkYwRjFGMEYwKiYtRio2JEklVTAzMkdGJkY5RjBJInhHRiZGMEYwLUYqNiRGNEY5RjBGJSEjPywmLUYqNiRGLEYuRjBGPSEjPQ==

unknowns(U1, U032, U031, U033);

NiZJI1UxRzYiSSVVMDMyR0YkSSVVMDMxR0YkSSVVMDMzR0Yk

run(Z);

<0> Put:

Ly1JI3BkRzYiNiRJJVUwMzFHRiVJInVHRiUsJEklVTAzMkdGJSIjPQ==

Ly1JI3BkRzYiNiRJJVUwMzFHRiVJJXVfeHhHRiUsJComSSJ1R0YlIiIiLUYkNiRJJVUwMzJHRiVGKEYsIiIn

Ly1JI3BkRzYiNiRJJVUwMzNHRiUqJEkldV94eEdGJSIiIywmKiYtRiQ2JEklVTAzMUdGJUYoIiIiSSJ0R0YlRjAhIiIqJi1GJDYkSSVVMDMyR0YlRihGMEkieEdGJUYwRjI=

Ly1JI3BkRzYiNiRJJVUwMzNHRiUqJkkidUdGJSIiIkkldV94eEdGJUYqLCQqJi1GJDYkSSVVMDMxR0YlRihGKkkidEdGJUYqISIi

Ly1JI3BkRzYiNiRJJVUwMzNHRiUqJkkldV94eEdGJSIiIkkidUdGJSIiIywkKiYtRiQ2JEklVTAzMUdGJUYoRipJInRHRiVGKiEiIg==

<0> Put:

Ly1JI3BkRzYiNiRJJVUwMzJHRiUqJEkldV94eEdGJSIiIyIiIQ==

Ly1JI3BkRzYiNiRJJVUwMzJHRiVJJXVfeHhHRiUiIiE=

Ly1JI3BkRzYiNiRJJVUwMzNHRiUqJEkidUdGJSIiJCIiIQ==

<0> Put:

Ly1JI3BkRzYiNiRJJVUwMzNHRiUqJEkidUdGJSIiIywkSSNVMUdGJSIjZw==

Ly1JI3BkRzYiNiRJJVUwMzNHRiVJJXVfeHhHRiUsJEkjVTFHRiUiIz8=

<0> Success!

dependence();

NiYvSSNVMUc2IjwiL0klVTAzMUdGJTwjSSJ1R0YlL0klVTAzMkdGJUYmL0klVTAzM0dGJTwkRipJJXVfeHhHRiU=

<Text-field style="Heading 3" italic="false" layout="Heading 3">Final step</Text-field>

Z := clear(pds);

NyUsJi1JI3BkRzYiNiRJJVUwMzFHRiZJInVHRiYiIiJJJVUwMzJHRiYhIz0sJkkjVTFHRiYhI2ctRiU2JEklVTAzM0dGJiokRikiIiNGKiwmLUYlNiRGMkkldV94eEdGJkYqRi4hIz8=

lprint(map(convert, Z, diff));

[diff(U031(u), u)-18*U032, -60*U1+diff(U033(u, u_xx), u, u), diff(U033(u, u_xx), u_xx)-20*U1]

U031 := 18*U032*u+C3; U033 := (20*u_xx+30*u^2)*U1+C1*u+C2; dependence(C1(), C2(),C3());

LCYqJkkidUc2IiIiIkklVTAzMkdGJUYmIiM9SSNDM0dGJUYm

LCgqJiwmSSV1X3h4RzYiIiM/KiRJInVHRiYiIiMiI0kiIiJJI1UxR0YmRixGLComSSNDMUdGJkYsRilGLEYsSSNDMkdGJkYs

NicvSSNDMUc2IjwiL0kjQzJHRiVGJi9JI0MzR0YlRiYvSSNVMUdGJUYmL0klVTAzMkdGJUYm

map(simpl,Z);

NyUiIiFGI0Yj

<Text-field style="Heading 3" italic="false" layout="Heading 3">The result</Text-field>

LCwqJkkjVTFHNiIiIiJJJXVfNXhHRiVGJkYmKiYsLComLCYqJkkidUdGJUYmSSVVMDMyR0YlRiYiIz1JI0MzR0YlRiZGJkkidEdGJUYmRiYqJkYuRiZJInhHRiVGJkYmKiYsJkkldV94eEdGJSIjPyokRi0iIiMiI0lGJkYkRiZGJiomSSNDMUdGJUYmRi1GJkYmSSNDMkdGJUYmRiZJJHVfeEdGJUYmRiYqJiwoKiZGLkYmRjFGJiIiJComRi1GJkYkRiYiIzVGPCNGJiIiJ0YmSSZ1X3h4eEdGJUYmRiZGLEY5RjBGRQ==

U := collect(U, [U1, C1, C2, C3, U032], factor);

LCwqJiwqSSV1XzV4RzYiIiIiKiZJInVHRiZGJ0kmdV94eHhHRiZGJyIjNSomSSR1X3hHRiZGJ0kldV94eEdGJkYnIiM/KiZGLUYnRikiIiMiI0lGJ0kjVTFHRiZGJ0YnKiYsJkYqI0YnIiInKiZGKUYnRi1GJ0YnRidJI0MxR0YmRidGJyomRi1GJ0kjQzJHRiZGJ0YnKiYsJiomRi1GJ0kidEdGJkYnRidGNkYnRidJI0MzR0YmRidGJyomLCoqKEYtRidGKUYnRj9GJyIjPSomRi1GJ0kieEdGJkYnRidGKUYxKiZGP0YnRipGJyIiJEYnSSVVMDMyR0YmRidGJw==