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A third order nonlinear ODE
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This ODE is reducible:
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![[[_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]](/support/helpjp/helpview.aspx?si=6705/file01309/math113.png)
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It can be solved by dsolve directly by determining an appropriate integrating factor (see odeadvisor,reducible), but let's consider a possible answer for it as a reduction of order from 3 to 1:
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![sol := y(x) = ODESolStruc(_c+Int(exp(Int(_f1(_c), _c)+_C1), _c)+_C2, [{diff(_f1(_c), _c) = 2*_f1(_c)^2+_f1(_c)*exp(_c)}, {_c = y(x)-x, _f1(_c) = -(diff(diff(y(x), x), x))/(-1+diff(y(x), x))^2}, {x = Int(exp(Int(_f1(_c), _c)+_C1), _c)+_C2, y(x) = _c+Int(exp(Int(_f1(_c), _c)+_C1), _c)+_C2}])](/support/helpjp/helpview.aspx?si=6705/file01309/math125.png)
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![sol := y(x) = `&where`(_c+Int(exp(Int(_f1(_c), _c)+_C1), _c)+_C2, [{diff(_f1(_c), _c) = 2*_f1(_c)^2+_f1(_c)*exp(_c)}, {_c = y(x)-x, _f1(_c) = -(diff(y(x), `$`(x, 2)))/(-1+diff(y(x), x))^2}, {x = Int(exp(Int(_f1(_c), _c)+_C1), _c)+_C2, y(x) = _c+Int(exp(Int(_f1(_c), _c)+_C1), _c)+_C2}])](/support/helpjp/helpview.aspx?si=6705/file01309/math128.png)
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Test that the above solves the ODE by using odetest:
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Now, the reduced ODE is of Bernoulli type, and can be selected using the mouse or through the following commands:
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From the above, it is clear that a particular solution to the reduced_ODE is given by
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from which a particular solution to the original ODE above can be built using
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In this "blackboard" example, dsolve succeeds in solving the reduced_ODE too, as follows:
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Passing this solution to buildsol, the general solution to ODE follows:
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| (12) |