(v) Number of arbitrary constants in the particular solution of a differential equation of order two is two. (

NCERT Exemplar · Q49

Q. (v) Number of arbitrary constants in the particular solution of a differential equation of order two is two. (State True or False.)

Puducherry CbseShort· 1mImportance★★★★★

Appeared in past exams:CBSE 2020· Set 65/1/1· 1mreworded

76% · 169/222 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Particular Solution Constants

The Intuition: Why "Particular" and Why "Constants"?

Imagine you're solving a puzzle where the answer isn't a single number, but a whole family of numbers. The equation $y' = 2x$ has infinitely many solutions: $y = x^2 + C$ , where $C$ can be any real number. Each different $C$ gives a different curve — all parallel parabolas shifted up or down.

Now suppose the puzzle gives you an extra condition: "the curve must pass through $(1, 3)$ ". That pins down exactly one member of the family: $3 = 1^2 + C \implies C = 2$ , so the answer is $y = x^2 + 2$ .

That $C = 2$ is a particular solution constant — the specific value of the arbitrary constant that makes the solution fit a given condition (an initial condition or boundary condition).

Note

The "general solution" contains arbitrary constants (like $C$ ). The "particular solution" has those constants replaced by specific numbers.

The Precise Statement

When solving a differential equation, you get a general solution of the form:

y = f(x, C_1, C_2, \dots, C_n)

where $C_1, C_2, \dots, C_n$ are arbitrary constants. The number of constants equals the order of the differential equation (first-order → one constant, second-order → two constants, etc.).

A particular solution is obtained by assigning specific numerical values to these constants. Those specific numbers are the particular solution constants, determined by imposing initial conditions or boundary conditions.

y = f(x, C_1, C_2, \dots, C_n)

A Concrete Example (First-Order)

Solve: $\frac{dy}{dx} = 3x^2$ , given $y(0) = 5$ .

Step 1 — General solution: $y = \int 3x^2 \, dx = x^3 + C$ .

Step 2 — Apply the condition: $y(0) = 5$ means when $x = 0$ , $y = 5$ : $5 = 0^3 + C \implies C = 5$ .

Step 3 — Particular solution: $y = x^3 + 5$ . The particular solution constant is $C = 5$ .

A Second-Order Example (Two Constants)

Solve: $\frac{d^2y}{dx^2} = 6x$ , given $y(0) = 1$ and $y'(0) = 2$ .

Step 1 — General solution: Integrate twice.

First integration: $y' = \int 6x \, dx = 3x^2 + C_1$

Second integration: $y = \int (3x^2 + C_1) \, dx = x^3 + C_1 x + C_2$

Step 2 — Apply conditions:

From $y'(0) = 2$ : $2 = 3(0)^2 + C_1 \implies C_1 = 2$

From $y(0) = 1$ : $1 = 0^3 + 2(0) + C_2 \implies C_2 = 1$ …

Start your 14-day free trial to unlock the full solution →

The statement is False. The particular solution of a differential equation has zero arbitrary constants — only the general solution contains arbitrary constants, and for a second-order equation, the general solution has two.

The key confusion here is between the general solution and the particular solution. Let’s be clear about what each means.

A differential equation of order two has a general solution that contains two arbitrary constants — these constants represent the family of all possible solutions. For example, the general solution of $\frac{d^2y}{dx^2} = 0$ is $y = C_1 x + C_2$ , where $C_1$ and $C_2$ are arbitrary.

A particular solution, on the other hand, is obtained when we assign specific numerical values to those constants using given initial or boundary conditions. Once the constants are fixed, they are no longer arbitrary — they become specific numbers. So a particular solution has zero arbitrary constants.

Watch out

A common mistake is to think that the number of arbitrary constants in the particular solution equals the order of the differential equation. That is true only for the general solution, not the particular solution.

Let’s walk through the reasoning step by step.

Order of a differential equation

The order is the highest derivative present. For a second-order equation, the general solution must contain two independent arbitrary constants — this is a fundamental theorem in differential equations.
General solution vs. particular solution
- General solution: Contains arbitrary constants (e.g., $y = A e^{x} + B e^{-x}$ for $y'' - y = 0$ ). …

Start your 14-day free trial to unlock the full solution →

←Previous questionQ48 — (iv) Correct substitution for the solution of the differential equatio…→Next questionQ50 — (vi) The differential equation representing the family of circles x²+(…

Q.(v) Number of arbitrary constants in the particular solution of a differential equation of order two is two. (State True or False.)

Concept understanding — Particular Solution Constants

The Intuition: Why "Particular" and Why "Constants"?

The Precise Statement

A Concrete Example (First-Order)

A Second-Order Example (Two Constants)

Method: Counting Arbitrary Constants — General vs Particular Solution

Steps

Common Mistakes

Q.(v) Number of arbitrary constants in the particular solution of a differential equation of order two is two. (State True or False.)

🎓Unlock everything free for 14 days

Unlock everything free for 14 days