Differential Equations

In this chapter: order and degree of a differential equation, general and particular solutions, forming a differential equation from a family of curves, and solving equations with variables separable, homogeneous equations and linear equations.

Order and degree

The order is the order of the highest derivative present. The degree is the power of that highest order derivative, and it is defined only when the equation is a polynomial in all the derivatives y′, y″, y‴, and so on.

In the second example, sin y does not affect the degree, because the polynomial condition applies to the derivatives, not to y itself. In the last one, sin(y′) is not a polynomial in y′, so the degree is not defined. Order and degree (when defined) are always positive integers.

General and particular solutions

A general solution contains arbitrary constants, and their number equals the order of the equation. A particular solution is obtained by giving the constants definite values, usually from an initial condition such as y(0) = 1. For example, y = A cos x + B sin x is the general solution of y″ + y = 0 (order 2, two constants), and y = cos x is the particular solution with y(0) = 1, y′(0) = 0.

Forming a differential equation

This topic was removed from the rationalised NCERT book, but it is still in the MHT‑CET and JEE Advanced syllabus. To form the differential equation of a family of curves with n arbitrary constants, differentiate n times and eliminate the constants. The result has order n. For the circles x² + y² = r² (one constant r), differentiating gives 2x + 2y y′ = 0, so the equation is x + y (dy/dx) = 0. For y = A cos x + B sin x (two constants), y″ = −y, so the equation is y″ + y = 0.

Variables separable

If the equation can be written as dy/dx = f(x) g(y), separate it as dy/g(y) = f(x) dx (for g(y) ≠ 0) and integrate both sides. Add a single constant on one side.

Worked example: Solve dy/dx = −x/y given that y = 4 when x = 3.
Solution: y dy = −x dx. Integrating, y²/2 = −x²/2 + c, that is, x² + y² = C. Using (3, 4): C = 9 + 16 = 25. The particular solution is x² + y² = 25, the highlighted circle in the figure.

Worked example: Solve dy/dx = e^{x + y}.
Solution: Write e^{x + y} = e^x e^y, so e^−y dy = e^x dx. Integrating, −e^−y = e^x + c, which can be written e^x + e^−y = C. Answers in MCQs are often in this rearranged form, so be ready to move the constant around.

Homogeneous differential equations

A function F(x, y) is homogeneous of degree n if F(λx, λy) = λⁿ F(x, y). The equation dy/dx = F(x, y) is homogeneous if F is homogeneous of degree zero, which means F can be written as a function of y/x alone. For example, (x² + y²)/(2xy) is homogeneous of degree zero.

Method: put y = vx, so dy/dx = v + x (dv/dx). The equation becomes separable in v and x. Solve, then replace v by y/x. (If the equation is in the form dx/dy = G(x, y), put x = vy instead.)

Worked example: Solve dy/dx = (x² + y²)/(2xy).
Solution: Put y = vx: v + x dv/dx = (1 + v²)/(2v). So x dv/dx = (1 + v² − 2v²)/(2v) = (1 − v²)/(2v). Separating, 2v dv/(1 − v²) = dx/x. Integrating, −log|1 − v²| = log|x| + c, which gives x(1 − v²) = C. Replacing v = y/x: x² − y² = Cx. Check by differentiating: 2x − 2yy′ = C = (x² − y²)/x, which rearranges to y′ = (x² + y²)/(2xy).

Linear differential equations

A first order linear equation has the form

If the equation is linear in x instead, dx/dy + P₁ x = Q₁ with P₁, Q₁ functions of y, the I.F. is e^{∫P₁ dy} and the solution is x × (I.F.) = ∫ Q₁ × (I.F.) dy + C. Always bring the equation to standard form (coefficient of dy/dx equal to 1) before reading off P.

Worked example: Solve x dy/dx + y = x³, x > 0.
Solution: Divide by x: dy/dx + (1/x) y = x². So P = 1/x, Q = x², and I.F. = e^∫dx/x = e^{log x} = x. Then y · x = ∫ x² · x dx = x⁴/4 + C, so y = x³/4 + C/x. Check: y′ + y/x = (3x²/4 − C/x²) + (x²/4 + C/x²) = x².

Useful integrating factors to recognise: P = tan x gives I.F. = sec x; P = cot x gives sin x; P = 2x gives e^x²; P = −1 gives e^−x.

Which method?

Some equations fit more than one type; any correct method gives an equivalent answer.

Common mistakes: (1) Stating the degree of an equation that is not a polynomial in its derivatives. (2) Forgetting the constant of integration, or adding constants on both sides and not combining them. (3) Reading P from an equation that is not in standard form, for example taking P = 1 in x dy/dx + y = x³. (4) Writing e^{log x} as log x instead of x. (5) After y = vx, forgetting to substitute back v = y/x in the final answer.

JEE and MHT‑CET focus

• Order and degree, including equations where the degree is not defined or needs rationalising first.
• Forming the differential equation of a given family of curves, and the number of arbitrary constants.
• Variables separable equations with initial conditions.
• Homogeneous equations by y = vx, and recognising them after simplification.
• Linear equations in y or in x, with standard integrating factors, and word problems on growth and decay (dy/dt = ky gives y = y₀e^kt).

Practice questions