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Mathematics and Statistics · Class 12 Commerce

Ch 8Differential Equation and Applications — Class 12 Mathematics and Statistics, concept-first.

A differential equation is an equation that connects an independent variable (say ), a dependent variable (say ) and one or more derivatives of with respect to . Because a derivative measures a rate of change, a differential equation is the natural language for any situation described by “the rate at which something ch…

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Key concepts

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Order and Degree of a Differential Equation

A differential equation is any equation that contains at least one derivative — ordinary or partial — of an unknown function.

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Chapter contents

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Differential Equation: Definition, Order and Degree

A differential equation is an equation that connects an independent variable (say ), a dependent variable (say ) and one or more derivatives of with respect to .

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Formation of a Differential Equation

A family of curves is described by an ordinary equation carrying one or more arbitrary constants (parameters).

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Solution by Variables Separable

A solution of a differential equation is a relation between and (free of derivatives) that satisfies the equation.

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Homogeneous Differential Equations

A first-order equation is homogeneous when it can be written so that depends on and only through their ratio : Equivalently, in the form , both and are homogeneous functions of the same degree (every…

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Linear Differential Equations of the First Order

A first-order linear differential equation is one in which and occur only to the first power and are not multiplied together — the standard form is where and are functions of alone (either may be cons…

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Applications: Growth, Decay and Population

Many real quantities change at a rate proportional to their current size. If is such a quantity at time , this law is written as the differential equation where is the constant of proportionality: des…

Exercises

+Show 7 questions7 questions
  1. Q11Find the order and degree of the differential equation $\displaystyle\frac{d^3y}{dx^3} + \left(\frac{d^2y}{dx^2}\right)^{4} + \frac{dy}{dx}…Free
  2. Q12Form the differential equation of the family $y = a x^{2} + b$, where $a$ and $b$ are arbitrary constants.Free
  3. Q13Solve $\displaystyle\frac{dy}{dx} = e^{x - y}$.Free
  4. Q14Solve the homogeneous equation $\displaystyle x\frac{dy}{dx} = x + y$.Preview
  5. Q15Solve the linear differential equation $\displaystyle\frac{dy}{dx} + y = x$.Preview
  6. Q16A bacterial culture grows at a rate proportional to the number present. Initially there are $1000$ bacteria, and after $1$ hour there are $2…Preview
  7. Q17Choose the correct option. The order and degree of the differential equation $\displaystyle\left(\frac{d^2y}{dx^2}\right)^{3} + \left(\frac{…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 23 questions23 questions
  1. Q1The order and degree of the differential equation $\left[1 + \left(\dfrac{dy}{dx}\right)^3\right]^{2/3} = 8\left(\dfrac{d^3y}{dx^3}\right)$…Preview
  2. Q2State whether the following statement is true or false. The integrating factor of the differential equation $\dfrac{dy}{dx} + \dfrac{y}{x} =…Preview
  3. Q3$y^2 = (x + c)^3$ is the general solution of the differential equation ______.Preview
  4. Q4Solve the following differential equation $x^2y\, dx - (x^3 + y^3)\, dy = 0$Preview
  5. Q5In a certain culture of bacteria, the rate of increase is proportional to the number present. If it is found that the number doubles in 4 ho…Preview
  6. Q6The integrating factor of $\frac{dy}{dx} + y = e^{-x}$ is ______. (a) $x$ (b) $-x$ (c) $e^x$ (d) $e^{-x}$Preview
  7. Q7The degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 = a^x$ is 3. (a) True (b) FalsePreview
  8. Q8For the differential equation, find the particular solution $(x - y^2 x) \, dx - (y + x^2 y) \, dy = 0$ when $x = 2$, $y = 0$Preview
  9. Q9In a certain culture of bacteria, the rate of increase is proportional to the number present. If it is found that the number doubles in 4 ho…Preview
  10. Q10The differential equation of $y = k_1 e^x + k_2 e^{-x}$ is ______. (a) $\dfrac{d^2y}{dx^2} - y = 0$ (b) $\dfrac{d^2y}{dx^2} + \dfrac{dy}{dx}…Preview
  11. Q11State whether the following statement is true or false: Order and degree of a differential equation are always positive integers. (a) True (…Preview
  12. Q12A solution of a differential equation which can be obtained from the general solution by giving particular values to the arbitrary constants…Preview
  13. Q13Find the differential equation whose general solution is $x^3 + y^3 = 35ax$.Preview
  14. Q14The rate of growth of population is proportional to the number present. If the population doubled in the last 25 years and the present popul…Preview
  15. Q15The order and degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = a^x$ are ______ respect…Preview
  16. Q16The integrating factor of the differential equation $\frac{dy}{dx} + \frac{y}{x} = x^3 - 3$ is ______. (a) $\log x$ (b) $e^x$ (c) $\frac{1}{…Preview
  17. Q17State whether the following statement is true or false: The differential equation obtained by eliminating arbitrary constants from $bx + ay…Preview
  18. Q18Obtain the differential equation by eliminating arbitrary constants from the following equation: $y = Ae^{3x} + Be^{-3x}$Preview
  19. Q19Solve the following differential equation $(x^2 - yx^2)\,dy + (y^2 + xy^2)\,dx = 0$. Separating the variables, the given equation can be wri…Preview
  20. Q20The solution of $\frac{dy}{dx} = 1$ is ______. (a) $x + y = c$ (b) $xy = c$ (c) $x^2 + y^2 = c$ (d) $y - x = c$Preview
  21. Q21State whether the following statement is true or false: Order and degree of a differential equation are always positive integers. (a) True (…Preview
  22. Q22Solve the following differential equation: $(x^2 - y^2)dx + 2xy\,dy = 0$Preview
  23. Q23Obtain the differential equation from the relation $Ax^2 + By^2 = 1$, where A and B are constants. Solution: The given equation is $Ax^2 + B…Preview

More questions

+Show 10 questions10 questions
  1. Example 1Determine the order and degree of the differential equation $\displaystyle\left(\frac{d^2y}{dx^2}\right)^{2} + \left(\frac{dy}{dx}\right)^{3…Free
  2. Example 2Find the order and degree of $\displaystyle\sqrt{1 + \left(\frac{dy}{dx}\right)^{2}} = \frac{d^2y}{dx^2}$.Free
  3. Example 3Form the differential equation of the family of curves $y = A e^{x} + B e^{-x}$, where $A$ and $B$ are arbitrary constants.Free
  4. Example 4Solve $\displaystyle\frac{dy}{dx} = \frac{1 + y^{2}}{1 + x^{2}}$.Preview
  5. Example 5Solve $\displaystyle\frac{dy}{dx} = 2xy$ given that $y = 1$ when $x = 0$.Preview
  6. Example 6Solve the homogeneous differential equation $\displaystyle\frac{dy}{dx} = \frac{x^{2} + y^{2}}{2xy}$.Preview
  7. Example 7Solve the linear differential equation $\displaystyle\frac{dy}{dx} + \frac{y}{x} = x^{2}$.Preview
  8. Example 8Solve $\displaystyle\frac{dy}{dx} + 2y = e^{3x}$.Preview
  9. Example 9The population of a town increases at a rate proportional to its current population. If the population doubles every $25$ years and the pres…Preview
  10. Example 10The value of a machine depreciates continuously at a rate proportional to its value. It was purchased for $\text{₹}1{,}00{,}000$ and after $…Preview