### Other calculators # Differential equation dx/dt=3xt^2

y() =
y'() =
y''() =
y'''() =
y''''() =

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### The solution

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d             2
--(x(t)) = 3*t *x(t)
dt                  
$$\frac{d}{d t} x{\left(t \right)} = 3 t^{2} x{\left(t \right)}$$
x' = 3*t^2*x
Detail solution
Given the equation:
$$\frac{d}{d t} x{\left(t \right)} = 3 t^{2} x{\left(t \right)}$$
This differential equation has the form:
y' + P(x)y = 0,

where
$$P{\left(t \right)} = - 3 t^{2}$$
and
and it is called linear homogeneous
differential first-order equation:
It's an equation with multiple variables.
The equation is solved using following steps:
From y' + P(x)y = 0 you get

$$\frac{dy}{y} = - P{\left(x \right)} dx$$, if y is not equal to 0
$$\int \frac{1}{y}\, dy = - \int P{\left(x \right)}\, dx$$
$$\log{\left(\left|{y}\right| \right)} = - \int P{\left(x \right)}\, dx$$
Or,
$$\left|{y}\right| = e^{- \int P{\left(x \right)}\, dx}$$
Therefore,
$$y_{1} = e^{- \int P{\left(x \right)}\, dx}$$
$$y_{2} = - e^{- \int P{\left(x \right)}\, dx}$$
The expression indicates that it is necessary to find the integral:
$$\int P{\left(x \right)}\, dx$$
Because
$$P{\left(t \right)} = - 3 t^{2}$$, then
$$\int P{\left(x \right)}\, dx$$ =
= $$\int \left(- 3 t^{2}\right)\, dt = - t^{3} + Const$$
Detailed solution of the integral
So, solution of the homogeneous linear equation:
$$y_{1} = e^{C_{1} + t^{3}}$$
$$y_{2} = - e^{C_{2} + t^{3}}$$
that leads to the correspondent solution
for any constant C, not equal to zero:
$$y = C e^{t^{3}}$$
           / 3\
\t /
x(t) = C1*e    
$$x{\left(t \right)} = C_{1} e^{t^{3}}$$
Graph of the Cauchy problem
The classification
separable
1st exact
1st linear
Bernoulli
almost linear
1st power series
lie group
separable Integral
1st exact Integral
1st linear Integral
Bernoulli Integral
almost linear Integral
(t, x):
(-10.0, 0.75)
(-7.777777777777778, 6.734200738135525e+32)
(-5.555555555555555, 2.17e-322)
(-3.333333333333333, nan)
(-1.1111111111111107, 2.78363573e-315)
(1.1111111111111107, 8.427456047434801e+197)
(3.333333333333334, 3.1933833808213433e-248)
(5.555555555555557, 6.234735095871012e-38)
(7.777777777777779, 8.388243566958251e+296)
(10.0, 3.861029683e-315)
(10.0, 3.861029683e-315)
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