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Business Mathematics and Statistics · Class 12 Commerce

Ch 4Differential Equations — Class 12 Business Mathematics and Statistics, concept-first.

This Tamil Nadu HSC Class 12 Business Mathematics and Statistics chapter introduces differential equations — equations involving an unknown function and its derivatives — building on the integration toolkit from the previous two chapters to actually SOLVE them.

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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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Definition, Order and Degree of a Differential Equation

This Tamil Nadu HSC Class 12 Business Mathematics and Statistics chapter introduces differential equations — equations involving an unknown function and its derivatives — building on the integration t…

2

Formation of a Differential Equation

A family of curves with arbitrary constants (like , one constant ) satisfies a differential equation of order , formed by differentiating the family's equation times and then eliminating the constant(…

3

Solving Differential Equations — the Variable-Separable Method

A first-order differential equation is variable-separable if it can be rearranged so that ALL the -terms (including ) are on one side, and ALL the -terms (including ) are on the other:

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Linear Differential Equations of the First Order

A first-order linear differential equation has the standard form

Exercises

+Show 4 questions4 questions
  1. Q8Find the order and degree of $\dfrac{d^3y}{dx^3}+x\left(\dfrac{dy}{dx}\right)^2-y=0$.Free
  2. Q9Form the differential equation representing the family of curves $y=A\cos x+B\sin x$, where $A,B$ are arbitrary constants.Free
  3. Q10Solve $(1+x)\,dy=(1+y)\,dx$.Preview
  4. Q11Solve $\dfrac{dy}{dx}+\dfrac{2}{x}y=x^2$ (for $x>0$).Preview

Sample & Board Papers

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

+Show 32 questions32 questions
  1. Q1The order and degree of the differential equation $\sqrt{\dfrac{d^2y}{dx^2}} = \sqrt{\dfrac{dy}{dx}} + 5$ are respectively : (a) $2$ and $3$…Preview
  2. Q2The P.I of $(3D^2 + D - 14)y = 13e^{2x}$ is : (a) $xe^{2x}$ (b) $\dfrac{x}{2}e^{2x}$ (c) $13xe^{2x}$ (d) $\dfrac{x^2}{2}e^{2x}$Preview
  3. Q3Find the differential equation of the family of all straight lines passing through the origin.Preview
  4. Q4The sum of $\text{₹ } 2{,}000$ is compounded continuously, the nominal rate of interest being $5\%$ per annum. In how many years, will the a…Preview
  5. Q5(a) Suppose that the quantity demanded $Q_d = 29 - 2p - 5\dfrac{dp}{dt} + \dfrac{d^2p}{dt^2}$ and quantity supplied $Q_s = 5 + 4p$, where $'…Preview
  6. Q6The order and degree of the differential equation $\left[\frac{d^2y}{dx^2}\right]^{\frac{3}{2}} - \sqrt{\left(\frac{dy}{dx}\right)} - 4 = 0$…Preview
  7. Q7A homogeneous differential equation of the form $\frac{dx}{dy} = f\left(\frac{x}{y}\right)$ can be solved by making substitution. (a) $y = v…Preview
  8. Q8Find the differential equation of the family of all straight lines passing through the origin.Preview
  9. Q9Solve : $(1 - x)dy - (1 + y)dx = 0$.Preview
  10. Q10(a) The slope of the tangent to a curve at any point $(x, y)$ on it is given by $(y^3 - 2yx^2)dx + (2xy^2 - x^3)dy = 0$ and the curve passes…Preview
  11. Q11The differential equation formed by eliminating A and B from $y=e^{-2x}(A\cos x+B\sin x)$ is : (a) $y_2-4y_1-5=0$ (b) $y_2-4y_1+5=0$ (c) $y_…Preview
  12. Q12The Particular Integral of $(3D^2+D-14)y=13e^{2x}$ is : (a) $\dfrac{x^2}{2}e^{2x}$ (b) $\dfrac{x}{2}e^{2x}$ (c) $13xe^{2x}$ (d) $xe^{2x}$Preview
  13. Q13Solve :- $\dfrac{dy}{dx}=xy+x+y+1$Preview
  14. Q14Solve : $9y''-12y'+4y=0$Preview
  15. Q15(a) Solve : $(D^2-2D+1)y=e^{2x}+e^x$. OR (b) An ambulance service claims that it takes on an average 8.9 minutes to reach its destination in…Preview
  16. Q16The order and degree of the differential equation $\sqrt{\frac{d^2y}{dx^2}}=\sqrt{\frac{dy}{dx}+5}$ are respectively : (a) 2 and 1 (b) 2 and…Preview
  17. Q17The general solution of the differential equation $\frac{dy}{dx}=\cos x$ is : (a) $y=\sin x-2$ (b) $y=\cos x+c$, $c$ is an arbitrary constan…Preview
  18. Q18Solve : $\log\left(\frac{dy}{dx}\right)=ax+by$Preview
  19. Q19Solve : $\frac{dy}{dx}+y\tan x=\cos^3 x$Preview
  20. Q20(a) Solve : $(y^2-2xy)dx=(x^2-2xy)dy$ OR (b) Construct Fisher's price index number and prove that it satisfies both Time Reversal Test and F…Preview
  21. Q21(a) Solve : $(3D^2+D-14)y=4-13e^{-\frac{7}{3}x}$ OR (b) If 18% of the bolts produced by a machine are defective, determine the probability t…Preview
  22. Q22The differential equation $\left(\dfrac{dx}{dy}\right)^{3}+2y^{1/2}=x$ is : (a) of order 1 and degree 6 (b) of order 2 and degree 1 (c) of o…Preview
  23. Q23The integrating factor of $x\dfrac{dy}{dx}-y=x^{2}$ is : (a) $\log x$ (b) $\dfrac{-1}{x}$ (c) $x$ (d) $\dfrac{1}{x}$Preview
  24. Q24Solve : $(D^{2}+2D+2)y=0$Preview
  25. Q25Find the differential equation of the family of curves $y=\dfrac{a}{x}+b$ where $a$ and $b$ are arbitrary constants.Preview
  26. Q26(a) If $\dfrac{dy}{dx}+2y\tan x=\sin x$ and if $y=0$ when $x=\dfrac{\pi}{3}$, express $y$ in terms of $x$. OR (b) Obtain an initial basic fe…Preview
  27. Q27(a) Solve the following differential equation. $\dfrac{dy}{dx}=\dfrac{x^{2}+y^{2}}{xy}$ OR (b) Out of 750 families with 4 children each, how…Preview
  28. Q28The complementary function of $(D^2 + 4)y = e^{2x}$ is : (a) $A\cos 2x + B\sin 2x$ (b) $(Ax + B)e^{2x}$ (c) $Ae^{-2x} + Be^{2x}$ (d) $(Ax +…Preview
  29. Q29The general solution of the differential equation $\frac{dy}{dx} = \cos x$ is : (a) $y = \cos x + c$, c is an arbitrary constant (b) $y = \s…Preview
  30. Q30Find the differential equation of the curve $x^2 + y^2 = a^2$Preview
  31. Q31Solve $(e^y + 1)\cos x \, dx + e^y \sin x \, dy = 0$Preview
  32. Q32(a) Solve : $(D^2 - 10D + 25)y = 4e^{5x} + 5$. OR (b) Calculate the Laspeyre's, Paasche's and Fisher's price index numbers for the following…Preview

More questions

+Show 7 questions7 questions
  1. Example 1Find the order and degree of $\dfrac{d^2y}{dx^2}+3\left(\dfrac{dy}{dx}\right)^2+5y=0$.Free
  2. Example 2Find the order and degree of $\left(\dfrac{d^2y}{dx^2}\right)^3+\left(\dfrac{dy}{dx}\right)^2=x$.Free
  3. Example 3Form the differential equation representing the family of curves $y=Ax^2$, where $A$ is an arbitrary constant.Free
  4. Example 4Form the differential equation representing the family of curves $y=Ae^{2x}$, where $A$ is an arbitrary constant.Preview
  5. Example 5Solve $\dfrac{dy}{dx}=\dfrac{y}{x}$.Preview
  6. Example 6Solve $\dfrac{dy}{dx}=e^{x+y}$.Preview
  7. Example 7Solve $\dfrac{dy}{dx}+y=e^x$.Preview