Mister Exam
Other calculators
- Linear homogeneous differential equations of 2nd order Step-By-Step
- Linear inhomogeneous differential equations of the 1st order Step-By-Step
- Differential equations with separable variables Step-by-Step
- A simplest differential equations of 1-order Step-by-Step
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How to use it?
- Differential equation:
- Equation dx/dt=3xt^2
- Equation dy/dx+y^2sinx=0
- Equation y"-7y=0
- Equation xy'-2y=x^3cosx
- Derivative of:
- dx/dt
- Integral of d{x}:
- 3xt^2
-
Identical expressions
- dx/dt=3xt^ two
- dx divide by dt equally 3xt squared
- dx divide by dt equally 3xt to the power of two
- dx/dt=3xt2
- dx/dt=3xt²
- dx/dt=3xt to the power of 2
- dx divide by dt=3xt^2
-
Expressions with functions
- dx
- dx/dt+3x/20+t=4
- dx/dt=-k(a-x)
- dx/dt=x
- dx/(xz-y)=dy/(yz-x)=dz/(xy-z)
- dx+e^(3x)dy=0
- dt
- dt*t*x/dx=t^2-1
- dt=5x-3
- dt=dy+csc(y)dt
- dt*(3*t+y4)+4*dy*t*y^3=0
- dt*(e^t+t^2*y)*y-dy*e^t=0
Differential equation dx/dt=3xt^2
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The solution
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[src]
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:
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:
$$\frac{dy}{y} = - P{\left(x \right)} dx$$, 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:
$$y = C e^{t^{3}}$$
$$\frac{d}{d t} x{\left(t \right)} = 3 t^{2} x{\left(t \right)}$$
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:
$$y = C e^{t^{3}}$$
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
Numerical answer
[src]
(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)
The graph