Is it correct to say: Taking degree as the highest power of ANY derivative?

Degree is the power of the HIGHEST-ORDER derivative only. In y‴ + 2(y″)² + y′ = 0 the degree is 1, not 2.

Is it correct to say: Stating a degree for equations like y″ + sin(y′) = 0?

The equation is not a polynomial in its derivatives, so the degree is not defined. The order (2 here) still exists.

Is it correct to say: Writing log|y| = x² + C ⇒ y = e^(x²) + C?

Exponentiating gives |y| = e^(x² + C) = e^C · e^(x²), so y = A e^(x²). The constant multiplies; it is not added.

Is it correct to say: Reading P before making the coefficient of dy/dx equal to 1?

For x dy/dx + 2y = x², divide by x first: dy/dx + (2/x)y = x. So P = 2/x and I.F. = e^(2 log x) = x².

Is it correct to say: Writing dy/dx = dv/dx after putting y = vx?

By the product rule, dy/dx = v + x dv/dx. Dropping 'v +' gives a completely different equation.

Is it correct to say: Substituting the initial condition before integrating?

Find the general solution with its constant first, then use the condition to fix C.

📄 Formula sheet (PDF)

Differential Equations

Order and degree, general and particular solutions, and solving first-order first-degree equations by separating variables, the homogeneous substitution y = vx and the integrating factor — NCERT Class 12 Maths Ch 9

📐 16 formulas✏️ 4 examples🎯 6 practice⚖️ Unit III: 35 marks (with Ch 5, 6, 7 and 8)🏫 CBSE📚 Class 12✓ 2025–26 syllabus

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Board Exam Tips

→Identify the type before solving: variables separable, homogeneous (F(λx, λy) = F(x, y)), or linear (dy/dx + Py = Q). Write the type in your answer.
→For a linear equation, divide through so that the coefficient of dy/dx is 1 BEFORE reading off P. Skipping this step is an easy way to get a wrong integrating factor.
→Simplify the integrating factor with e^(log f(x)) = f(x). For example, e^(2 log x) = x², not 2x.
→For a particular solution, find the general solution first, then substitute the given point to find C. State the final equation clearly.
→Degree is defined only when the equation is a polynomial in its derivatives. If you see sin(y′), e^(y′) or similar, the degree is not defined.
→Add the constant of integration once, when you integrate, and carry it through the algebra.

📐 Formulas(16)

Order of a Differential Equation

Order = order of the highest derivative present

Degree of a Differential Equation★ Board fav

x (\frac{d ^{2} y}{d x ^{2}})^{3} + (\frac{d y}{d x})^{4} - y = 0 \Rightarrow order 2, degree 3

When Degree is Not Defined

\frac{d ^{2} y}{d x ^{2}} + sin (\frac{d y}{d x}) = 0 \Rightarrow degree not defined

General and Particular Solutions

order n \Rightarrow general solution has n arbitrary constants

Particular Solution from a Condition

y (x_{0}) = y_{0} \Rightarrow substitute in the general solution to find C

Variables Separable★ Board fav

\frac{d y}{d x} = g (x) h (y) \Rightarrow \int \frac{d y}{h ( y )} = \int g (x) d x + C

Symbol	Meaning
$g (x)$	Factor depending on x only
$h (y)$	Factor depending on y only
$C$	Arbitrary constant

Homogeneous Function of Degree n

F (λ x, λ y) = λ^{n} F (x, y)

Symbol	Meaning
$λ$	Any non-zero constant
$n$	Degree of homogeneity

Homogeneous Differential Equation

\frac{d y}{d x} = F (x, y), F homogeneous of degree 0 \Leftrightarrow \frac{d y}{d x} = g (\frac{y}{x})

Substitution y = vx★ Board fav

y = v x \Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}

Homogeneous Equation after Substitution

v + x \frac{d v}{d x} = g (v) \Rightarrow \int \frac{d v}{g ( v ) - v} = \int \frac{d x}{x} + C

Substitution x = vy

\frac{d x}{d y} = h (\frac{x}{y}) : x = v y \Rightarrow \frac{d x}{d y} = v + y \frac{d v}{d y}

Linear Differential Equation

\frac{d y}{d x} + P y = Q, P, Q constants or functions of x only

Symbol	Meaning
$P$	Coefficient of y (function of x or constant)
$Q$	Right-hand side (function of x or constant)

Integrating Factor★ Board fav

I.F. = e^{\int P d x}

Solution of a Linear Equation★ Board fav

y \cdot (I.F.) = \int Q \cdot (I.F.) d x + C

Linear Equation in x

\frac{d x}{d y} + P_{1} x = Q_{1} : I.F. = e^{\int P_{1} d y}, x \cdot (I.F.) = \int Q_{1} \cdot (I.F.) d y + C

Useful Simplifications of the I.F.

e^{l o g x} = x, e^{- l o g x} = \frac{1}{x}, e^{k l o g x} = x^{k} (x > 0)

✏️ Solved Examples

1Solved Exampleeasy2 steps

Find the order and degree (if defined) of (i) x(y″)³ + (y′)⁴ − y = 0 and (ii) y″ + e^(y′) = 0.

(i) The highest derivative is y″, so the order is 2. The equation is a polynomial in y″ and y′, and the power of y″ is 3.

Order 2, degree 3

2Solved Exampleboard4 steps

Find the particular solution of dy/dx = 2x(1 + y²), given that y = 1 when x = 0.

The right side is a product of a function of x and a function of y, so separate the variables.

\frac{d y}{1 + y ^{2}} = 2 x d x

3Solved Exampleboard5 steps

Solve xy (dy/dx) = x² + y², given that y = 2 when x = 1.

Write dy/dx = F(x, y) = (x² + y²)/(xy). Then F(λx, λy) = λ²(x² + y²)/(λ²xy) = F(x, y), so the equation is homogeneous of degree 0.

\frac{d y}{d x} = \frac{x ^{2} + y ^{2}}{x y} = \frac{x}{y} + \frac{y}{x}

4Solved ExampleHOTS5 steps

The slope of the tangent to a curve at any point (x, y) is y + 2x. If the curve passes through (0, 1), find its equation.

Slope = dy/dx, so dy/dx = y + 2x. Rearrange into linear form.

\frac{d y}{d x} - y = 2 x (P = - 1, Q = 2 x)

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Taking degree as the highest power of ANY derivative
✓Degree is the power of the HIGHEST-ORDER derivative only. In y‴ + 2(y″)² + y′ = 0 the degree is 1, not 2.
2
Stating a degree for equations like y″ + sin(y′) = 0
✓The equation is not a polynomial in its derivatives, so the degree is not defined. The order (2 here) still exists.
3
Writing log|y| = x² + C ⇒ y = e^(x²) + C
✓Exponentiating gives |y| = e^(x² + C) = e^C · e^(x²), so y = A e^(x²). The constant multiplies; it is not added.
4
Reading P before making the coefficient of dy/dx equal to 1
✓For x dy/dx + 2y = x², divide by x first: dy/dx + (2/x)y = x. So P = 2/x and I.F. = e^(2 log x) = x².
5
Writing dy/dx = dv/dx after putting y = vx
✓By the product rule, dy/dx = v + x dv/dx. Dropping 'v +' gives a completely different equation.
6
Substituting the initial condition before integrating
✓Find the general solution with its constant first, then use the condition to fix C.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Find the order and degree of y‴ + 2(y″)² + y′ = 0.
Q2
Find the general solution of dy/dx = (1 + y)/(1 + x).
Q3
Find the particular solution of dy/dx = 3x²y, given that y = 2 when x = 0.
Q4
Solve the homogeneous equation dy/dx = y/x + sec(y/x).
Q5
Solve dy/dx + 2y = 6eˣ.
Q6
Solve x dy/dx + y = x³, x > 0.

📝 Notes

Differential Equations

A differential equation relates an unknown function to its derivatives. Chapter 9 covers how to classify these equations (order and degree) and three methods for solving first-order, first-degree equations.

Classify first

Order: the highest derivative that appears.
Degree: the power of that highest derivative, but only if the equation is a polynomial in its derivatives. Otherwise, write "not defined".
General vs particular: an equation of order n has n arbitrary constants in its general solution. A particular solution has none.

Which method to use

Variables separable. If dy/dx can be written as g(x)·h(y), separate the variables and integrate both sides.
Homogeneous. If replacing x, y by λx, λy leaves dy/dx unchanged, substitute y = vx (or x = vy) and the equation becomes separable in v.
Linear. If the equation can be written as dy/dx + Py = Q with P, Q functions of x only, use I.F. = e^(∫P dx) and y·(I.F.) = ∫Q·(I.F.) dx + C. If it is linear in x instead, swap the roles of x and y.

Check in this order. Some equations fit more than one type, and any correct method leads to an equivalent solution, possibly written in a different form.

Writing a full-mark solution

State the type and, for homogeneous equations, show the F(λx, λy) test.
For linear equations, write P, Q and the I.F. on separate lines.
Integrate both sides once, adding a single constant C.
For a particular solution, substitute the given values only at the end, and box the final equation.