The Unknown Function
Given:
$$\int_0^x f(t)dt + xf(x) = x^2$$
Solution:
Let $F(x) = \int_0^x f(t)dt$.
Then, the given equation can be written as
$$F(x) + xf(x) = x^2$$
Differentiating both sides with respect to $x$ yields
$$f(x) + F'(x) + xf'(x) = 2x$$
Substituting $F'(x)$ with $f(x)$ gives
$f(x) + xf'(x) = 2x$$
Rearranging the equation yields
$$f'(x) + \frac{2}{x}f(x) = 2$$
This is a firstorder linear differential equation with the general solution
$$f(x) = c_1e^{\frac{2}{x}} + \frac{2x}{e^{\frac{2}{x}}}$$
where $c_1$ is an arbitrary constant.
Integrodifferential equation
Differential equations  

Navier–Stokes differential equations used to simulate airflow around an obstruction


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General topics


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In mathematics, an integrodifferential equation is an equation that involves both integrals and derivatives of a function.
General first order linear equations
The general firstorder, linear (only with respect to the term involving derivative) integrodifferential equation is of the form

$$
d
d
x
u
(
x
)
+
∫
x
0
x
f
(
t
,
u
(
t
)
)
d
t
=
g
(
x
,
u
(
x
)
)
,
u
(
x
0
)
=
u
0
,
x
0
≥
0.
{\displaystyle \frac ddxu(x)+\int _x_0^xf(t,u
differential equations, obtaining a closedform solution can often be difficult. In the relatively few cases where a solution can be found, it is often by some kind of integral transform, where the problem is first transformed into an algebraic setting. In such situations, the solution of the problem may be derived by applying the inverse transform to the solution of this algebraic equation.
Example
Consider the following secondorder problem,

$$
u
′
(
x
)
+
2
u
(
x
)
+
5
∫
0
x
u
(
t
)
d
t
=
θ
(
x
)
with
u
(
0
)
=
0
,
{\displaystyle u'(x)+2u(x)+5\int _0^xu
θ
(
x
)
=
{
1
,
x
≥
0
0
,
x
<
0
{\displaystyle \theta (x)=\left\\beginarrayll1,\qquad x\geq 0\\0,\qquad x<0\endarray\right.
is the Heaviside step function. The Laplace transform is defined by,

$$
U
(
s
)
=
L
u
(
x
)
=
∫
0
∞
e
−
s
x
u
(
x
)
d
x
.
\displaystyle U(s)=\mathcal L\left\u(x)\right\=\int _0^\infty e^sxu(x)\,dx.
Upon taking termbyterm Laplace transforms, and utilising the rules for derivatives and integrals, the integrodifferential equation is converted into the following algebraic equation,

$$
s
U
(
s
)
−
u
(
0
)
+
2
U
(
s
)
+
5
s
U
(
s
)
=
1
s
.
\displaystyle sU(s)u(0)+2U(s)+\frac 5sU(s)=\frac 1s.
Thus,

$$
U
(
s
)
=
1
s
2
+
2
s
+
5
\displaystyle U(s)=\frac 1s^2+2s+5
.
Inverting the Laplace transform using contour integral methods then gives

$$
u
(
x
)
=
1
2
e
−
x
sin
(
2
x
)
θ
(
x
)
\displaystyle u(x)=\frac 12e^x\sin(2x)\theta (x)
.
Alternatively, one can complete the square and use a table of Laplace transforms (“exponentially decaying sine wave”) or recall from memory to proceed:

$$
U
(
s
)
=
1
s
2
+
2
s
+
5
=
1
2
2
(
s
+
1
)
2
+
4
⇒
u
(
x
)
=
L
−
1
U
(
s
)
=
1
2
e
−
x
sin
(
2
x
)
θ
(
x
)
\displaystyle U(s)=\frac 1s^2+2s+5=\frac 12\frac 2(s+1)^2+4\Rightarrow u(x)=\mathcal L^1\left\U(s)\right\=\frac 12e^x\sin(2x)\theta (x)
.
Applications
Integrodifferential equations model many situations from science and engineering, such as in circuit analysis. By Kirchhoff’s second law, the net voltage drop across a closed loop equals the voltage impressed
$$
E
(
t
)
{\displaystyle E
L
d
d
t
I
(
t
)
+
R
I
(
t
)
+
1
C
∫
0
t
I
(
τ
)
d
τ
=
E
(
t
)
,
{\displaystyle L\frac ddtI

$$
is the resistance,
$$the inductance, and
$$the capacitance.
The activity of interacting inhibitory and excitatory neurons can be described by a system of integrodifferential equations, see for example the WilsonCowan model.
Epidemiology
Integrodifferential equations have found applications in epidemiology, the mathematical modeling of epidemics, particularly when the models contain agestructure or describe spatial epidemics.
See also
References
Further reading
 Vangipuram Lakshmikantham, M. Rama Mohana Rao, “Theory of IntegroDifferential Equations”, CRC Press, 1995
External links
 Interactive Mathematics
 Numerical solution of the example using Chebfun
Classification 



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Applications  
Mathematicians 
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