Chapter Exam

Unit 8: Differential Equations — Online MCQ Test

Q: What is the order of the differential equation \(\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - 5y = 0\)?

Answer: B. 2 — The highest derivative present is \(\frac{d^2y}{dx^2}\), so the order is 2.

Q: Which of the following is a differential equation?

Answer: B. \(\frac{dy}{dx} = x + y\) — A differential equation contains derivatives of the dependent variable. Only option B includes a derivative.

Q: Solve the separable differential equation \(\frac{dy}{dx} = xy\).

Answer: A. \(y = Ce^{x^2/2}\) — Separating variables gives \(\frac{dy}{y} = x\,dx\). Integrating, \(\ln|y| = \frac{x^2}{2} + C\), hence \(y = Ce^{x^2/2}\).

Q: If the differential equation is \(\frac{dy}{dx} = \frac{x+y}{x}\), then after substituting \(y = vx\), the equation becomes

Answer: B. \(x\frac{dv}{dx} = 1\) — Using \(y=vx\), we get \(\frac{dy}{dx}=v+x\frac{dv}{dx}\). Substituting gives \(v+x\frac{dv}{dx}=1+v\), so \(x\frac{dv}{dx}=1\).

Q: If a first-order differential equation has integrating factor \(e^{\int 2x\,dx}\), then the corresponding coefficient \(P(x)\) is

Answer: B. 2x — The integrating factor for \(\frac{dy}{dx} + P(x)y = Q(x)\) is \(e^{\int P(x)dx}\). Hence \(P(x)=2x\).

MATHEMATICS - 2B · CLASS 12 INTERMEDIATE 2 YEAR · Telangana State Board

Practice Unit 8: Differential Equations with a free chapter-wise online MCQ test for Telangana State Board CLASS 12 INTERMEDIATE 2 YEAR MATHEMATICS - 2B. This chapter covers: The final chapter details order degree of differential equations formation of differential equations solutions of variable separable homogeneous and linear first-order differential.... AI-generated questions from basic to board-exam level, with instant results and explanations.

Questions

20m

Time Limit

Attempts Left

10 random questions from this chapter (mixed difficulty)
Questions you've seen before won't repeat until the pool resets
You have 20 minutes — exam auto-submits when time is up
Maximum 3 attempts per chapter
Results and explanations shown immediately after submission

New here? Register free — includes 3 free chapter exams.

Unit 8: Differential Equations — Important Questions & Answers (FAQ)

Frequently asked questions from Telangana State Board CLASS 12 INTERMEDIATE 2 YEAR MATHEMATICS - 2B — Unit 8: Differential Equations, with answers and explanations. These are sample questions; the exam has its own separate question set.

What is the order of the differential equation \(\frac{d^2y}{dx^2} + 3\frac{dy}{dx} - 5y = 0\)?

A. 1
B. 2 ✓
C. 3
D. 5

Answer: B. 2
The highest derivative present is \(\frac{d^2y}{dx^2}\), so the order is 2.

Which of the following is a differential equation?

A. \(y = x^2 + 1\)
B. \(\frac{dy}{dx} = x + y\) ✓
C. \(x^2 + y^2 = 1\)
D. \(y = \sin x\)

Answer: B. \(\frac{dy}{dx} = x + y\)
A differential equation contains derivatives of the dependent variable. Only option B includes a derivative.

Solve the separable differential equation \(\frac{dy}{dx} = xy\).

A. \(y = Ce^{x^2/2}\) ✓
B. \(y = Ce^{2x}\)
C. \(y = Cx^2\)
D. \(y = C\ln x\)

Answer: A. \(y = Ce^{x^2/2}\)
Separating variables gives \(\frac{dy}{y} = x\,dx\). Integrating, \(\ln|y| = \frac{x^2}{2} + C\), hence \(y = Ce^{x^2/2}\).

If the differential equation is \(\frac{dy}{dx} = \frac{x+y}{x}\), then after substituting \(y = vx\), the equation becomes

A. \(v + x\frac{dv}{dx} = 1 + v\)
B. \(x\frac{dv}{dx} = 1\) ✓
C. \(x\frac{dv}{dx} = v\)
D. \(\frac{dv}{dx} = x+v\)

Answer: B. \(x\frac{dv}{dx} = 1\)
Using \(y=vx\), we get \(\frac{dy}{dx}=v+x\frac{dv}{dx}\). Substituting gives \(v+x\frac{dv}{dx}=1+v\), so \(x\frac{dv}{dx}=1\).

If a first-order differential equation has integrating factor \(e^{\int 2x\,dx}\), then the corresponding coefficient \(P(x)\) is