
Linear differential equations are a type of mathematical equation defined by a linear polynomial in the unknown function and its derivatives. The solutions of homogeneous linear differential equations with polynomial coefficients are called holonomic functions, which include many common functions such as the exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric functions.
The equation $y'' = -y$ is a linear differential equation, and its solutions are $y_1 = \cos(x)$ and $y_2 = \sin(x)$, which are linearly independent. Therefore, the general solution is $y(x) = c_1\sin(x) + c_2\cos(x)$, where $c_1$ and $c_2$ are arbitrary constants.
So, the solutions to the linear differential equation $y'' = -y$ are linear combinations of sine and cosine functions.
What You'll Learn
- Linear differential equations can have sin(y)
- Linear differential equations are defined by a linear equation in unknown variables and their derivatives
- A linear differential equation is one that has the form: dy/dx + p(x)y = q(x)
- The general solution to a linear first-order differential equation is: y(t) = (∫μ(t)g(t) dt + c)/μ(t)
- The integrating factor, μ(t), is defined as: μ(t) = e^(∫p(t) dt)
Linear differential equations can have sin(y)
A linear differential equation is one that has the following form:
> dy/dx + p(x)y = q(x)
In this equation, y is a function and dy/dx is a derivative.
A linear differential equation can also be defined as a linear polynomial equation, which consists of derivatives of several variables.
A linear differential equation can include sin(x), as in the following example:
> dy/dx + sin(x)y = 0
Here, p(x) = -sin(x) and q(x) = 0.
However, an equation in the form dy/dx = sin(y) is considered non-linear. This is because the dependent variable, y, is in a trigonometric function. For an equation to be linear, the variable must be on its own with a function of x as its prefactor in the non-differential term.
In conclusion, linear differential equations can include sin(x), but not sin(y).
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Linear differential equations are defined by a linear equation in unknown variables and their derivatives
A linear differential equation is defined by a
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A linear differential equation is one that has the form: dy/dx + p(x)y = q(x)
A linear differential equation is one that has the form:
\[
\frac{dy}{dx} + p(x)y = q(x)
\]
Where y is a function and dy/dx is a derivative. The solution of the linear differential equation produces the value of the variable y.
P and Q are either constants or functions of the independent variable (in this case, x) only.
The linear differential equation in y is of the form:
\[
\frac{dy}{dx} + py = q
\]
Where p and q are functions in x.
The linear differential equation in x is:
\[
\frac{dx}{dy} + p_1x = q_1
\]
Where p1 and q1 are functions of y.
An example of a linear differential equation in y is:
\[
\frac{dy}{dx} + y = \cos x
\]
An example of a linear differential equation in x is:
\[
\frac{dx}{dy} + x = \sin y
\]
A first-order differential equation is considered linear if it can be written in the form:
\[
\frac{dy}{dx} + p(x)y = q(x)
\]
If the dependent variable is in a trigonometric function, it is considered non-linear. For example, the equation:
\[
Y' = \sin y
\]
Cannot be written in the above form and is therefore non-linear.
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The general solution to a linear first-order differential equation is: y(t) = (∫μ(t)g(t) dt + c)/μ(t)
A linear differential equation is one that has the form:
> dy/dx + p(x)y = q(x)
Where p(x) and q(x) are either constants or functions of the independent variable (in this case, x).
A first-order linear differential equation is one where p(x) and q(x) are either constants or functions of y (the independent variable).
The general solution to a linear first-order differential equation is:
> y(t) = (∫μ(t)g(t) dt + c)/μ(t)
Where μ(t) is the integrating factor, which is found by taking the integral of p(x) with respect to x, and c is an arbitrary constant.
This solution can be derived as follows:
First, rearrange the equation to the form:
> dy/dx + Py = Q
Then, find the integrating factor, μ(x), by integrating P (obtained in the previous step) with respect to x and putting this integral as a power to e:
> μ(x) = e^∫Pdx
Next, multiply both sides of the equation by the integrating factor:
> μ(x) * (dy/dx + Py) = μ(x)Q
Simplify the left-hand side:
> d(y * μ(x))/dx = Q * μ(x)
Integrate both sides with respect to x:
> ∫d(y * μ(x)) = ∫Q * μ(x) dx
And finally, simplify and add the constant term:
> y * μ(x) = ∫Q * μ(x) dx + c
> y = (∫μ(t)g(t) dt + c)/μ(t)
It is important to note that linear differential equations cannot have transcendental functions of y (the dependent variable) or dy/dx. For example, the equation dy/dx + x sin y = lnx is not a linear differential equation because the second term, x sin y, is a transcendental function of y.
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The integrating factor, μ(t), is defined as: μ(t) = e^(∫p(t) dt)
A linear differential equation is one that has the form:
$$\frac{dy}{dx} + p(x)y = q(x)$$
Where $y$ is a function and $\frac{dy}{dx}$ is its derivative.
A linear differential equation can be solved by finding the integrating factor, $M(x)$, which is defined as:
$$M(x) = e^{\int p(x) dx}$$
The integrating factor is used to rewrite the differential equation in a form where the left-hand side is the derivative of $yM(x)$. This is done by multiplying both sides of the equation by $M(x)$:
$$M(x)\frac{dy}{dx} + M(x)p(x)y = M(x)q(x)$$
$$\frac{d(yM(x))}{dx} = M(x)q(x)$$
Integrating both sides with respect to $x$ then gives:
$$y = \frac{1}{M(x)} \left( \int M(x)q(x) dx + C \right)$$
Where $C$ is an arbitrary constant.
This method can be used to solve linear differential equations of the form:
$$\frac{dy}{dx} + Py = Q$$
Where $P$ and $Q$ are either constants or functions of the independent variable $x$.
However, it's important to note that not all differential equations are linear. A differential equation is considered non-linear if it cannot be written in the standard form of a linear differential equation. For example, the equation:
$$y' = \sin y$$
Cannot be written in the standard form and is therefore non-linear.
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Frequently asked questions
No, a linear differential equation cannot have sinusoidal functions or any other transcendental functions of the dependent variable. It can only have the dependent variable in its first power.
The general form of a first-order linear differential equation is: dy/dx + p(x)y = q(x), where p(x) and q(x) are functions of the independent variable x.
A linear differential equation has the dependent variable and its derivatives multiplied by constants or functions of the independent variable. A nonlinear differential equation does not follow this pattern and can have more complex expressions involving the dependent variable and its derivatives.
To solve a first-order linear differential equation, you can use an integrating factor, rearrange the equation to isolate the derivative, and then integrate both sides. This will give you the general solution, which you can then use to find the particular solution that satisfies any given initial conditions.
An example of a linear differential equation is: dy/dx + 2y = x^2. This equation has the dependent variable y and its derivative multiplied by constants, and there are no sinusoidal functions or higher powers of y.