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Find the general solution of the given differential equation. Give the largest over which the general solution is defined.

$\dfrac{dr}{d\theta}+r\sec(\theta)=\cos(\theta)$

This question aims to find the general solution of the given differential equation and interval in which the solution defines. When any constant of the general solution takes on some unique value, then the solution becomes a particular solution of the equation. By applying boundary conditions (also known as initial conditions), a particular solution to the differential equation is obtained. To obtain a particular solution, a general solution is first found, and then a particular solution is generated using the given conditions.

Suppose:

\[\dfrac{dy}{dx}=e^{x}+\cos(2x)\]

Thus, the general solution is given as follows:

\[y=e^{x}+\dfrac{\sin2x}{2}\]

A general solution of an nth-order differential equation involves $n$ necessary arbitrary constants. When we solve a first-order differential equation by the method of separable variables, we must necessarily introduce an arbitrary constant as soon as the integration is done. So you can see that the solution of the first-order differential equation has the necessary arbitrary constant after simplification.

Similarly, general solution of a second-order differential equation will contain the $2$ necessary arbitrary constants, and so on. The general solution geometrically represents an n-parameter family of curves. For example, general solution of the differential equation $\dfrac{dy}{dx}$$=4x^{3}$, which turns out to be $y$$=$$x^{4}$$+c$, where $c$ is an arbitrary constant.

Particular Solution

Particular solution of a differential equation is the solution obtained from the general solution by assigning particular values to arbitrary constants. The conditions for calculating the values of arbitrary constants can be given to us in the form of an initial value problem or boundary conditions depending on the problem.

Singular solution

The singular solution is also a particular solution of a given differential equation, but it cannot be obtained from general solution by specifying the values of arbitrary constants.

Expert Answer

The given equation is:

\[\dfrac{dr}{d\theta}+r\sec(\theta)=\cos(\theta)\]

\[Integrating\: factor=e^{\int\sec\theta d\theta}\]

\[=e^{\ln(\sec\theta+\tan\theta)}\]

\[=\sec\theta+\tan\theta\]

The solution is given by:

\[r(\sec\theta+\tan\theta)=\int(\sec\theta+\tan\theta)\cos\theta\theta+c\]

\[=\int(1+\sin\theta)d\theta+c\]

\[=\theta-\cos\theta+c\]

Hence, the general solution is given as follows:

\[r(\theta)=\dfrac{\theta}{\sec\theta+\tan\theta}-\dfrac{\cos\theta}{\sec\theta+\tan\theta}+\dfrac{c}{\sec\theta+\tan\theta}\]

The largest interval for which the solution is defined.

The solution doesn’t exist for the $\sec\theta+\tan\theta=0$.

  1. $\sec\theta$ is defined for all real numbers except integral multiple of $\dfrac{\pi}{2}$.
  2. $\tan\theta$ is defined for all real numbers except integral multiple of $\dfrac{\pi}{2}$.

Thus, $\sec\theta+\tan\theta$ is defined for all the real numbers except $\dfrac{\pi}{2}$.

Hence, the largest interval of existence is $(-\dfrac{\pi}{2},\dfrac{\pi}{2})$.

Numerical Result

The general solution for the differential equation is given as follows:

\[r(\theta)=\dfrac{\theta}{\sec\theta+\tan\theta}-\dfrac{\cos\theta}{\sec\theta+\tan\theta}+\dfrac{c}{\sec\theta+\tan\theta}\]

The largest interval of existence for the $\sec\theta+\tan\theta$ is $(-\dfrac{\pi}{2},\dfrac{\pi}{2})$.

Example

Find the general solution of given differential equation. $x^{2}\dfrac{dy}{dx} + xy = 8$. It gives the largest interval on which the general solution is defined.

Solution

Given, $x^{2}\dfrac{dy}{dx}+x.y=8$

\[x^{2}+\dfrac{dy}{dx}+x.y=8\]

Divide both sides by $x^{2}$.

\[\dfrac{dy}{dx}+\dfrac{y}{x}=\dfrac{8}{x^{2}}\]

Equation can be written in the form,$\dfrac{dy}{dx}+A(x)y=B(x)$ is the linear differential equation where $A(x)=\dfrac{1}{x}$ and $B(x)=\dfrac{8}{x^{2}}$.

\[Integrating\:factor=e^{\int A(x)dx}\]

\[I.F=e^{\int \dfrac{1}{x}.dx}\]

\[=e^{log_{e}x}\]

\[=x\]

Solution of a linear differential equation is given by:

\[xy=\int x.(\dfrac{8}{x^{2}})dx\]

\[xy=8\dfrac{1}{x}dx\]

\[xy=8\log_{e}x+C\]

This general solution is defined as $∀$ $x$ $ϵ$ $R$ $+$ because if $x = 0$ or $x = -ve$, the $\log_{e}x$ does not exist.

Solution of the linear differential equation is:

\[xy=8\log_{e}x+C\]

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