Mister Exam

Differential equation dy/dx=4y

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

from to

The solution

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d
--(y(x)) = 4*y(x)
dx               
$$\frac{d}{d x} y{\left(x \right)} = 4 y{\left(x \right)}$$
y' = 4*y
Detail solution

Step

Given the equation:
$$\frac{d}{d x} y{\left(x \right)} = 4 y{\left(x \right)}$$
This differential equation has the form:
$$y' + P(x)y = 0$$,
where
$$P{\left(x \right)} = -4$$
and
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
$$\int \frac{1}{y}\, dy = - \int P{\left(x \right)}\, dx$$
$$\log{\left(\left|{y}\right| \right)} = - \int P{\left(x \right)}\, dx$$
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
$$P{\left(x \right)} = -4$$, then
$$\int P{\left(x \right)}\, dx = \int \left(-4\right)\, dx = - 4 x + Const$$
Detailed solution of the integral
So, solution of the homogeneous linear equation:
$$y_{1} = e^{C_{1} + 4 x}$$
$$y_{2} = - e^{C_{2} + 4 x}$$
that leads to the correspondent solution for any constant C, not equal to zero:
$$y = C e^{4 x}$$
           4*x
y(x) = C1*e   
$$y{\left(x \right)} = C_{1} e^{4 x}$$
$$y\left(x\right)=y\left(0\right)\,e^{4\,x}$$
y = y(0)*E^(4*x)
Graph of the Cauchy problem
The classification
1st exact
1st exact Integral
1st linear
1st linear Integral
1st power series
Bernoulli
Bernoulli Integral
almost linear
almost linear Integral
lie group
nth linear constant coeff homogeneous
separable
separable Integral
(x, y):
(-10.0, 0.75)
(-7.777777777777778, nan)
(-5.555555555555555, nan)
(-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
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