CBSE Class 12 Mathematics Chapter 9: Differential Equations NCERT Solutions

The given differential equation is $\backslash frac\{d^4y\}\{dx^4\}\; +\; \backslash sin(y\text{'}\text{'}\text{'})\; =\; 0$. We can rewrite $\backslash frac\{d^4y\}\{dx^4\}$ as $y\text{'}\text{'}\text{'}\text{'}$ and $y\text{'}\text{'}\text{'}$ as the third derivative of y with respect to x.

The highest order derivative present in the equation is $y\text{'}\text{'}\text{'}\text{'}$, which is the fourth derivative. Therefore, the order of the differential equation is 4.

To determine the degree, we check if the equation is a polynomial in its derivatives. The term $\backslash sin(y\text{'}\text{'}\text{'})$ involves the sine of a derivative, which means the equation is not a polynomial in its derivatives. Hence, the degree of this differential equation is not defined.

The given differential equation is $y\text{'}\; +\; 5y\; =\; 0$.

The highest order derivative present in the equation is $y\text{'}$, which is the first derivative. Therefore, the order of the differential equation is 1.

The equation is a polynomial in its derivatives. The highest power of the highest order derivative ($y\text{'}$) is 1. Thus, the degree of the differential equation is 1.

The given differential equation is $\backslash left(\backslash frac\{ds\}\{dt\}\backslash right)^4\; +\; 3s\backslash frac\{d^2s\}\{dt^2\}\; =\; 0$.

The derivatives present are $\backslash frac\{ds\}\{dt\}$ (first derivative) and $\backslash frac\{d^2s\}\{dt^2\}$ (second derivative). The highest order derivative is $\backslash frac\{d^2s\}\{dt^2\}$. Therefore, the order of the differential equation is 2.

The equation is a polynomial in its derivatives ($\backslash frac\{ds\}\{dt\}$ and $\backslash frac\{d^2s\}\{dt^2\}$). The highest power of the highest order derivative ($\backslash frac\{d^2s\}\{dt^2\}$) is 1. Thus, the degree of the differential equation is 1.

The given differential equation is $\backslash left(\backslash frac\{d^2y\}\{dx^2\}\backslash right)^2\; +\; \backslash cos\backslash left(\backslash frac\{dy\}\{dx\}\backslash right)\; =\; 0$.

The highest order derivative present is $\backslash frac\{d^2y\}\{dx^2\}$, which is the second derivative. Therefore, the order of the differential equation is 2.

To determine the degree, we examine the powers of the derivatives. However, the term $\backslash cos\backslash left(\backslash frac\{dy\}\{dx\}\backslash right)$ involves the cosine of the first derivative. Since the equation is not a polynomial in its derivatives, the degree is not defined.

The given differential equation is $\backslash left(\backslash frac\{d^2y\}\{dx^2\}\backslash right)^2\; =\; \backslash cos\; 3x\; +\; \backslash sin\; 3x$.

We can rewrite this as $\backslash left(\backslash frac\{d^2y\}\{dx^2\}\backslash right)^2\; -\; \backslash cos\; 3x\; -\; \backslash sin\; 3x\; =\; 0$.

The highest order derivative present is $\backslash frac\{d^2y\}\{dx^2\}$, which is the second derivative. Therefore, the order of the differential equation is 2.

The equation is a polynomial in the derivative $\backslash frac\{d^2y\}\{dx^2\}$. The highest power of this highest order derivative is 2. Thus, the degree of the differential equation is 2.

The given differential equation is $(y\text{'}\text{'}\text{'})^2\; +\; (y\text{'}\text{'})^3\; +\; (y\text{'})^4\; +\; y^5\; =\; 0$.

The derivatives present are $y\text{'}$, $y\text{'}\text{'}$, and $y\text{'}\text{'}\text{'}$. The highest order derivative is $y\text{'}\text{'}\text{'}$, which is the third derivative. Therefore, the order of the differential equation is 3.

The equation is a polynomial in its derivatives ($y\text{'}\text{'}\text{'}$, $y\text{'}\text{'}$, and $y\text{'}$). The highest power of the highest order derivative ($y\text{'}\text{'}\text{'}$) is 2. Thus, the degree of the differential equation is 2.