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Solve the differential equation $x^{\prime}=-2x+4e^{-t}$

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Solution

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Final answer to the problem

$x=e^{-2t}\left(4e^t+C_0\right)$
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Step-by-step Solution

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  • Exact Differential Equation
  • Linear Differential Equation
  • Separable Differential Equation
  • Homogeneous Differential Equation
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
  • Integrate by substitution
  • Integrate by parts
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1

Rewrite the differential equation using Leibniz notation

$\frac{dx}{dt}=-2x+4e^{-t}$

Learn how to solve differential equations problems step by step online.

$\frac{dx}{dt}=-2x+4e^{-t}$

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Unlock the first 3 steps of this solution

Learn how to solve differential equations problems step by step online. Solve the differential equation x^'=-2x+4e^(-t). Rewrite the differential equation using Leibniz notation. Rearrange the differential equation. We can identify that the differential equation has the form: \frac{dy}{dx} + P(x)\cdot y(x) = Q(x), so we can classify it as a linear first order differential equation, where P(t)=2 and Q(t)=4e^{-t}. In order to solve the differential equation, the first step is to find the integrating factor \mu(x). To find \mu(t), we first need to calculate \int P(t)dt.

Final answer to the problem

$x=e^{-2t}\left(4e^t+C_0\right)$

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Function Plot

Plotting: $x=e^{-2t}\left(4e^t+C_0\right)$

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0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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Main Topic: Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives.

Used Formulas

See formulas (3)

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