Business Mathematics and Basic Statistics · Class 12 Commerce
Ch 15Differential Equations — Class 12 Business Mathematics and Basic Statistics, concept-first.
An ordinary differential equation is an equation that connects an unknown function of a single independent variable with one or more of its derivatives — , , and so on. For example, , or , or are all differential equations.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
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.
Most relevant Q&A
- Find the order and degree of the differential equation $\left(\dfrac{d^3y}{dx^3}\right)^2 + \left(\dfrac{dy}{dx}\right)^5 = 0$.Free
- Find the order and degree of the differential equation $\left(\dfrac{d^2y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^2 = x$.Free
- Find the order and degree of the differential equation $\sqrt{1+\left(\dfrac{dy}{dx}\right)^2} = x\dfrac{d^2y}{dx^2}$.Free
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Definition of a Differential Equation
An ordinary differential equation is an equation that connects an unknown function of a single independent variable with one or more of its derivatives — , , and so on.
Order and Degree of a Differential Equation
Two simple numbers describe the “shape” of a differential equation before any attempt is made to solve it: its order and its degree.
Formation of a Differential Equation for Simple Cases
“Forming” a differential equation reverses the idea of a general solution: starting from a family of curves given by an equation containing one or more arbitrary constants, the task is to find a diffe…
Solving by the Method of Variable Separation
The variable-separable method solves a differential equation of the special form , where the right side can be written as a function of alone multiplied (or divided) by a function of alone.
Verifying a Solution of a Differential Equation
Whether a solution was reached by variable separation (§4) or was simply given as a candidate function to check, the same direct test confirms whether it genuinely satisfies a differential equation: d…
Exercises
+−Show 5 questionsHide questions5 questions
- Q8Find the order and degree of the differential equation $\left(\dfrac{d^3y}{dx^3}\right)^2 + \left(\dfrac{dy}{dx}\right)^5 = 0$.Free
- Q9Form the differential equation representing the family of curves $x^2 + y^2 = c$, where $c$ is an arbitrary positive constant (concentric ci…Free
- Q10Solve the differential equation $\dfrac{dy}{dx} = \dfrac{y^2}{x^2}$ by the method of variable separation.Preview
- Q11Solve the differential equation $x\,dy + y\,dx = 0$ (for $x, y \neq 0$), expressing the solution as a relation between $x$ and $y$.Preview
- Q12Verify that $y^2 = 4ax$, where $a$ is an arbitrary constant, satisfies the differential equation $2x\dfrac{dy}{dx} = y$ (i.e., show the cons…Preview
More questions
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- Example 1Find the order and degree of the differential equation $\left(\dfrac{d^2y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^2 = x$.Free
- Example 2Find the order and degree of the differential equation $\sqrt{1+\left(\dfrac{dy}{dx}\right)^2} = x\dfrac{d^2y}{dx^2}$.Free
- Example 3Form the differential equation representing the family of curves $y = cx^2$, where $c$ is an arbitrary constant.Free
- Example 4Form the differential equation representing the family of curves $y = Ax + Bx^2$, where $A$ and $B$ are arbitrary constants.Preview
- Example 5Solve the differential equation $\dfrac{dy}{dx} = \dfrac{x^2}{y^2}$ by the method of variable separation.Preview
- Example 6Solve the differential equation $\dfrac{dy}{dx} = \dfrac{y}{x}$ (for $x, y > 0$) by the method of variable separation.Preview
- Example 7Verify that $y = kx$, where $k$ is an arbitrary constant, is a solution of the differential equation $x\dfrac{dy}{dx} = y$.Preview