Class 12 — Mathematics (NCERT)

Differential Equations

Class 12

✓By the end of this lesson students will be able to define the order and degree of a differential equation.
✓By the end of this lesson students will be able to solve first-order, first-degree differential equations using the variable separable method.
✓By the end of this lesson students will be able to identify and solve first-order linear differential equations.
✓By the end of this lesson students will be able to apply appropriate methods to solve various types of differential equations.

Key concepts

Differential Equation

An equation involving independent variable(s), dependent variable(s) and the derivatives of the dependent variable(s) with respect to the independent variable(s) is called a differential equation. For example, dy/dx = cos x, d²y/dx² + y = 0.

Order of a Differential Equation

The order of a differential equation is the order of the highest order derivative appearing in the differential equation. For example, the order of d²y/dx² + (dy/dx)³ + y = 0 is 2.

Degree of a Differential Equation

The degree of a differential equation, when it is a polynomial equation in derivatives, is the highest power (positive integral index) of the highest order derivative involved in the equation. If the differential equation is not a polynomial equation in its derivatives (e.g., involves sin(dy/dx), e^(dy/dx), log(dy/dx)), then its degree is not defined. For example, the degree of d²y/dx² + (dy/dx)³ + y = 0 is 1 (as the highest order derivative d²y/dx² has power 1).

Variable Separable Method

A first-order, first-degree differential equation of the form dy/dx = f(x, y) can be solved by the method of separation of variables if f(x, y) can be expressed as a product g(x)h(y). The equation can then be written as dy/h(y) = g(x)dx. Integrating both sides with respect to their respective variables yields the general solution.

∫(1/h(y)) dy = ∫g(x) dx + C

Linear Differential Equation (First Order)

A first-order differential equation is called a linear differential equation if it is of the form dy/dx + Py = Q, where P and Q are functions of x or constants. Alternatively, it can be of the form dx/dy + P₁x = Q₁, where P₁ and Q₁ are functions of y or constants.

General Solution for dy/dx + Py = Q: y ⋅ (I.F.) = ∫(Q ⋅ I.F.) dx + C, where I.F. (Integrating Factor) = e^(∫P dx)

Key facts to remember

1A differential equation involves derivatives of a dependent variable with respect to an independent variable.
2The order of a differential equation is the order of the highest derivative present.
3The degree of a differential equation is the power of the highest order derivative, provided the equation is a polynomial in derivatives.
4If a differential equation is not a polynomial in its derivatives, its degree is not defined.
5The variable separable method is applicable when the function f(x, y) in dy/dx = f(x, y) can be expressed as a product g(x)h(y).
6A first-order linear differential equation is of the form dy/dx + Py = Q, where P and Q are functions of x or constants.
7The Integrating Factor (I.F.) for dy/dx + Py = Q is e^(∫P dx).
8The general solution for a linear differential equation dy/dx + Py = Q is y ⋅ (I.F.) = ∫(Q ⋅ I.F.) dx + C.

Worked examples

Example 1

Find the order and degree of the differential equation: (d³y/dx³) + (d²y/dx²)² + (dy/dx)⁵ = 0

IThe given differential equation is (d³y/dx³) + (d²y/dx²)² + (dy/dx)⁵ = 0.

IIThe highest order derivative present in the equation is d³y/dx³.

IIITherefore, the order of the differential equation is 3.

IVThe equation is a polynomial in derivatives. The power of the highest order derivative (d³y/dx³) is 1.

VTherefore, the degree of the differential equation is 1.

Answer

Order = 3, Degree = 1

Example 2

Solve the differential equation: dy/dx = (1+y²)/(1+x²)

IThe given differential equation is dy/dx = (1+y²)/(1+x²).

IIWe can separate the variables by rearranging the terms:

IIIdy/(1+y²) = dx/(1+x²)

IVNow, integrate both sides:

V∫ dy/(1+y²) = ∫ dx/(1+x²)

VIUsing the standard integral formula ∫ dx/(a²+x²) = (1/a) tan⁻¹(x/a), we get:

VIItan⁻¹y = tan⁻¹x + C

VIIIThis is the general solution.

Answer

tan⁻¹y = tan⁻¹x + C

Remember to add the constant of integration 'C' after integrating.

Example 3

Solve the differential equation: x(dy/dx) + 2y = x² (x ≠ 0)

IThe given differential equation is x(dy/dx) + 2y = x².

IITo convert it into the standard form dy/dx + Py = Q, divide the entire equation by x (since x ≠ 0):

IIIdy/dx + (2/x)y = x

IVComparing this with dy/dx + Py = Q, we have P = 2/x and Q = x.

VNow, calculate the Integrating Factor (I.F.):

VII.F. = e^(∫P dx) = e^(∫(2/x) dx) = e^(2 log|x|) = e^(log|x|²) = x² (assuming x > 0 for simplicity, or |x|² = x²).

VIIThe general solution is given by y ⋅ (I.F.) = ∫(Q ⋅ I.F.) dx + C.

VIIISubstitute the values:

9y ⋅ x² = ∫(x ⋅ x²) dx + C

10y ⋅ x² = ∫x³ dx + C

11y ⋅ x² = (x⁴/4) + C

12To express y explicitly, divide by x²:

13y = (x⁴/4x²) + (C/x²)

14y = (x²/4) + (C/x²)

Answer

y = (x²/4) + (C/x²)

Always ensure the linear differential equation is in the standard form dy/dx + Py = Q before identifying P and Q.

Common mistakes

✗Incorrectly determining the degree of a differential equation when it is not a polynomial in its derivatives (e.g., trying to find the degree of sin(dy/dx) + y = 0).
✗Errors in integration, especially with trigonometric or logarithmic functions, during the variable separable method.
✗Forgetting to add the constant of integration 'C' after performing indefinite integration.
✗Not writing the linear differential equation in the standard form dy/dx + Py = Q before identifying P and Q, leading to incorrect P and Q values.
✗Mistakes in calculating the integrating factor, particularly with signs or exponents.

Exam tips

★Always check if the differential equation is a polynomial in its derivatives before stating its degree. If not, state that the degree is 'not defined'.
★Practice various integration techniques thoroughly, as they are crucial for solving differential equations.
★Clearly identify P and Q when solving linear differential equations to avoid errors in calculating the integrating factor.
★Show all steps clearly and systematically in your solutions, especially for variable separable and linear differential equations, to ensure full marks.

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