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

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 $\sec\theta+\tan\theta=0$.

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

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