Mister Exam

# Differential equation dx/dt+3x/20+t=4

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

from to

### The solution

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    3*x(t)   d
t + ------ + --(x(t)) = 4
20     dt          
$$t + \frac{3 x{\left(t \right)}}{20} + \frac{d}{d t} x{\left(t \right)} = 4$$
t + 3*x/20 + x' = 4
Detail solution
Given the equation:
$$t + \frac{3 x{\left(t \right)}}{20} + \frac{d}{d t} x{\left(t \right)} = 4$$
This differential equation has the form:
y' + P(x)y = Q(x)

where
$$P{\left(t \right)} = \frac{3}{20}$$
and
$$Q{\left(t \right)} = - t + 4$$
and it is called linear inhomogeneous
differential first-order equation:
First of all, we should solve the correspondent linear homogeneous equation
y' + P(x)y = 0

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
$$P{\left(t \right)} = \frac{3}{20}$$, then
$$\int P{\left(x \right)}\, dx = \int \frac{3}{20}\, dt = \frac{3 t}{20} + Const$$
Detailed solution of the integral
So, solution of the homogeneous linear equation:
$$y_{1} = e^{C_{1} - \frac{3 t}{20}}$$
$$y_{2} = - e^{C_{2} - \frac{3 t}{20}}$$
that leads to the correspondent solution
for any constant C, not equal to zero:
$$y = C e^{- \frac{3 t}{20}}$$
We get a solution for the correspondent homogeneous equation
Now we should solve the inhomogeneous equation
y' + P(x)y = Q(x)

Use variation of parameters method
Now, consider C a function of x

$$y = C{\left(t \right)} e^{- \frac{3 t}{20}}$$
And apply it in the original equation.
Using the rules:
- for product differentiation;
- of composite functions derivative,
we find that
$$\frac{d}{d x} C{\left(x \right)} = Q{\left(x \right)} e^{\int P{\left(x \right)}\, dx}$$
Let use Q(x) and P(x) for this equation.
We get the first-order differential equation for C(x):
$$\frac{d}{d t} C{\left(t \right)} = \left(- t + 4\right) e^{\frac{3 t}{20}}$$
So,
$$C{\left(t \right)} = \int \left(- t + 4\right) e^{\frac{3 t}{20}}\, dt = - \frac{\left(60 t - 400\right) e^{\frac{3 t}{20}}}{9} + \frac{80 e^{\frac{3 t}{20}}}{3} + Const$$
Detailed solution of the integral
use C(x) at
$$y = C{\left(t \right)} e^{- \frac{3 t}{20}}$$
and we get a definitive solution for y(x):
$$e^{- \frac{3 t}{20}} \left(- \frac{\left(60 t - 400\right) e^{\frac{3 t}{20}}}{9} + \frac{80 e^{\frac{3 t}{20}}}{3} + Const\right)$$
                        -3*t
----
640   20*t        20
x(t) = --- - ---- + C1*e
9     3             
$$x{\left(t \right)} = - \frac{20 t}{3} + \frac{640}{9} + C_{1} e^{- \frac{3 t}{20}}$$
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 undetermined coefficients
nth linear constant coeff variation of parameters
nth linear constant coeff variation of parameters Integral
(t, x):
(-10.0, 0.75)
(-7.777777777777778, 24.778269808045504)
(-5.555555555555555, 37.795741403184614)
(-3.333333333333333, 42.92363099671872)
(-1.1111111111111107, 42.39838841788831)
(1.1111111111111107, 37.82249939227275)
(3.333333333333334, 30.344195499666775)
(5.555555555555557, 20.786220529235543)
(7.777777777777779, 9.738096082942961)
(10.0, -2.377767114898042)
(10.0, -2.377767114898042)
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