RD Sharma Solutions Class 12 Differential Equations

RD Sharma Solutions Class 12 Chapter 22

An equation containing a dependent variable, an independent variable and differential coefficients of dependent variable w.r.t. the independent variable in known as a differential Equation.

General Form of a Differential Equation:

\(\boldsymbol{P_{0}\left ( \frac{d^{n}y}{dx^{n}} \right )\;+\;P_{1}\left ( \frac{d^{n-1}y}{dx^{n-1}} \right )\;+\;P_{2}\left ( \frac{d^{n-2}y}{dx^{n-2}} \right )\;+ \;. \;.\; . \;+ \;P_{n-1}\frac{dy}{dx}+P_{n}y\;=\;Q }\)

Examples:

\(\frac{dx}{dy}\;=\;5yx\;,\;\frac{dy}{dx}\;=\;2x^{3} + 12xy\;,\; \frac{dx}{dy}-sin \;x=cos\;x\;,\;\frac{d^{2y}}{dx^{2}}-5x=0\)

Order of a Differential Equation:

It is the order of the highest order derivative appearing in the equation.

Examples: The Equation \(\frac{d^{2}y}{dx^{2}}= 5y+8 \left ( \frac{dy}{dx} \right )^{2}\) is of order 3

The equation \(\frac{d^{5}y}{dx^{5}}= -6y+15 \left ( \frac{dy}{dx} \right )^{7}\) is of order 5.

Degree of a Differential Equation:

It is the degree of the highest order derivative (Differential coefficients are free from fraction and radicals).

Examples: The Equation \(\frac{d^{4}y}{dx^{4}}-7y= 10 \left ( \frac{dy}{dx} \right )^{8}\) is of degree 1 because, the power of the highest order derivative in this differential equation is 1.

The Equation \(\left ( \frac{d^{3}y}{dx^{3}} \right )^{5}= 21y-10 \left ( \frac{dy}{dx} \right )^{2}\) is of degree 5 because, the power of the highest order derivative in this differential equation is 5.

RD Sharma Solutions for Class 12 maths Differential Equations will help you to understand the different forms of differential Equations, its Applications in solving real life problems like Newton’s law of cooling, problems related to growth and decay, coordinate geometry and more in a very simplified way. Understand all these concepts in detail with complete exercise wise solved RD Sharma Solutions to answer complex problems asked in highly competitive exams like JEE Mains and Advanced accurately.

Practise This Question

On decreasing the temperature, what happens to the solubility of solute in solution?

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