Mister Exam

Differential equation dx/dt=x

For Cauchy problem:

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

The graph:

from to

The solution

You have entered [src]
--(x(t)) = x(t)
$$\frac{d}{d t} x{\left(t \right)} = x{\left(t \right)}$$
x' = x
Detail solution


Given the equation:
$$\frac{d}{d t} x{\left(t \right)} = x{\left(t \right)}$$
This differential equation has the form:
$$y' + P(x)y = 0$$,
$$P{\left(t \right)} = -1$$
and it is called linear homogeneous differential first-order equation:
It's the equation with separable 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
$$\int \frac{1}{y}\, dy = - \int P{\left(x \right)}\, dx$$
$$\log{\left(\left|{y}\right| \right)} = - \int P{\left(x \right)}\, dx$$
$$\left|{y}\right| = e^{- \int P{\left(x \right)}\, dx}$$
$$y_{1} = e^{- \int P{\left(x \right)}\, dx}$$
$$y_{2} = - e^{- \int P{\left(x \right)}\, dx}$$
The expression indicates that it is necessary to find the integral:
$$\int P{\left(x \right)}\, dx$$
$$P{\left(t \right)} = -1$$, then
$$\int P{\left(x \right)}\, dx = \int \left(-1\right)\, dt = - t + Const$$
Detailed solution of the integral
So, solution of the homogeneous linear equation:
$$y_{1} = e^{C_{1} + t}$$
$$y_{2} = - e^{C_{2} + t}$$
that leads to the correspondent solution for any constant C, not equal to zero:
$$y = C e^{t}$$
The answer (#2) [src]
x = x(0)*E^t
The answer [src]
x(t) = C1*e 
$$x{\left(t \right)} = C_{1} e^{t}$$
Graph of the Cauchy problem
The classification
1st exact
1st exact Integral
1st linear
1st linear Integral
1st power series
Bernoulli Integral
almost linear
almost linear Integral
lie group
nth linear constant coeff homogeneous
separable Integral
Numerical answer [src]
(t, x):
(-10.0, 0.75)
(-7.777777777777778, 6.920861621442101)
(-5.555555555555555, 63.86442821962763)
(-3.333333333333333, nan)
(-1.1111111111111107, nan)
(1.1111111111111107, nan)
(3.333333333333334, nan)
(5.555555555555557, nan)
(7.777777777777779, nan)
(10.0, nan)
(10.0, nan)
The graph
Differential equation dx/dt=x
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