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CDS Differential Equations

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This page covers CDS Differential Equations with complete concept notes, 37 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Differential Equations— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic
Separable equations

Rearrange so all y terms are on one side and all x terms on the other, then integrate both sides. 2

Linear first-order equations

Use the standard form dy/dx + Py = Q, where P and Q are functions of x. 3

Homogeneous equations

Variables can be separated by substitution.

2.Formula Block

Memorise — at least one formula appears in every paper
Separable form: dy/dx = f(x)g(y)
Solution: ∫(1/g(y))dy = ∫f(x)dx + C
Linear first-order: dy/dx + Py = Q
Integrating Factor = e^(∫P dx)
Solution: y × IF = ∫Q × IF dx + C

3.Exam Patterns

What examiners ask — read before attempting PYQs

NDA asks mostly first-order differential equations. Common question types: find general solution, find particular solution given initial condition, identify order/degree, verify if a function is a solution. Shortcut/Trick: Always check if the equation is separable first—this is the quickest method. If variables can be separated, do it immediately before attempting other methods. For verification problems, just differentiate the given solution and check if it satisfies the original equation.

4.Worked Example

Solve this step-by-step before moving on
1
Step 1

Separate variables. dy = 2x dx

2
Step 2

Integrate both sides. ∫dy = ∫2x dx y = x² + C

3
Step 3

Apply initial condition y(0) = 1. 1 = 0² + C C = 1

4
Step 4

Write particular solution. y = x² + 1

5.Exam Traps

Common mistakes students make — avoid these

Students forget to add the constant of integration (C) in the general solution. Always include it. Also, they sometimes fail to simplify or rearrange before integrating, missing the separable structure.

Key Points to Remember

  • A differential equation connects a function with its derivatives and expresses how one quantity changes with respect to another.
  • Order of a differential equation is determined by the highest derivative present in it.
  • Separable equations can be solved by rearranging so variables are on opposite sides, then integrating each side separately.
  • For linear first-order equations (dy/dx + Py = Q), use integrating factor method: IF = e^(∫P dx).
  • Initial conditions (given values at specific points) help find the particular solution from the general solution.
  • Always verify your solution by differentiating it and checking whether it satisfies the original equation.

Exam-Specific Tips

  • Order of differential equation = highest power of the highest derivative present.
  • Degree of differential equation = power of the highest order derivative (only when equation is polynomial in derivatives).
  • Separable differential equation general form: dy/dx = f(x)g(y) can be solved as ∫(1/g(y))dy = ∫f(x)dx + C.
  • Integrating Factor for dy/dx + Py = Q is always e^(∫P dx), where P is function of x only.
  • A particular solution is obtained by substituting initial/boundary conditions into the general solution to find the constant C.
  • Homogeneous differential equation of first order solved using substitution y = vx, converting it to separable form.
  • dy/dx = k(constant) has general solution y = kx + C, representing straight line family.
  • Solution to dy/dx = ky is exponential: y = Ae^(kx), where A is determined by initial condition.
Practice MCQs

Differential Equations — Practice Questions

37graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 37

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Practice 1easy

The differential equation dy/dx = 2xy satisfies the initial condition y(0) = 1. What is y(x)?

Practice 2easy

The differential equation dy/dx = 2xy with initial condition y(0) = 1 has the solution:

Practice 3easy

The order and degree of the differential equation (d²y/dx²)² + (dy/dx)³ = x are respectively:

Practice 4easy

The solution to the differential equation dy/dx + y = e^(-x) is:

Practice 5easy

If y = e^(2x)·sin(3x), then d²y/dx² equals:

Practice 6easy

The differential equation (1 + x²)dy/dx + 2xy = 4x² is linear. Its solution is:

Practice 7easy

The order and degree of the differential equation d²y/dx² + (dy/dx)³ = sin(x) are respectively:

Practice 8easy

If y = e^(2x)·sin(3x), then dy/dx equals:

Practice 9easy

The solution of the differential equation (1 + x²)dy/dx = xy with y(0) = 2 is:

Practice 10easy

The differential equation d²y/dx² + 4y = 0 has the general solution y = A·cos(2x) + B·sin(2x). If y(0) = 3 and dy/dx|_(x=0) = 4, what is the value of A + B?

Practice 11easy

Consider the differential equation dy/dx = (y² + 1)/x. Which of the following is the correct form after separating variables?

Practice 12easy

The differential equation d²y/dx² - 5·dy/dx + 6y = 0 is a second-order linear homogeneous equation. What is the characteristic equation?

Practice 13easy

The solution to the differential equation dy/dx = 3e^(2x) with y(0) = 5 is:

Practice 14medium

The differential equation dy/dx = (y - x)/(y + x) can be solved by the substitution y = vx. After substitution and simplification, the resulting separable equation is:

Practice 15medium

The differential equation dy/dx = (y² + 2xy)/(x²) can be reduced to a linear form by the substitution v = y/x. After substitution, the resulting differential equation in v and x is:

Practice 16medium

The solution to the differential equation dy/dx + y·cot(x) = 2x·csc(x) with the condition y(π/2) = 1 is:

Practice 17medium

Consider the differential equation (1 + x²)dy/dx + 2xy = 4x². The general solution is:

Practice 18medium

The differential equation d²y/dx² - 5dy/dx + 6y = 0 has the general solution:

Practice 19medium

The solution to the differential equation dy/dx = (y - x)/(y + x) that passes through the point (1, 2) is:

Practice 20medium

The differential equation dy/dx = (y² + 2xy)/(x²) can be transformed into a linear differential equation by the substitution v = y/x. After substitution, the resulting differential equation in v and x is:

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60-Second Revision — Differential Equations

  • Remember: Order = highest derivative power; Degree = power of highest order derivative in polynomial form.
  • Formula: Separable equations → rearrange and integrate both sides separately, add C to general solution.
  • Formula: Linear first-order → use Integrating Factor = e^(∫P dx), then y×IF = ∫Q×IF dx + C.
  • Trap: Don't forget the constant C in general solution; verify your answer by differentiating it.
  • Exam trick: If equation is separable, always separate first—it's faster than any other method.
  • Initial conditions convert general solution to particular solution; substitute given point values to find C.
  • Quick check: Differentiate your proposed solution and substitute back into original equation to verify.
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