Differential Equations — Practice

Question 1

Determine the order and degree (if defined) of y''+2y'+sin y=0.

Accepted Answer

The highest derivative present is y'', so the order is 2. Since sin y is a function of y itself (not of a derivative), the equation is still polynomial in its derivatives. Answer: Order 2, degree 1.

Question 2

Find the general solution of dy/dx+y=1 (y is not 1).

Accepted Answer

Separating gives dy/(1-y)=dx, and integrating gives -log|1-y|=x+C1. Answer: y=1-Ce^(-x).

Question 3

Show that y'=(x+y)/x is homogeneous and solve it.

Accepted Answer

With y=vx, the equation becomes v+x(dv/dx)=1+v, so x(dv/dx)=1, which integrates directly to v=log|x|+C. Answer: y=x(log|x|+C).

Question 4

Find the general solution of dy/dx+y/x=x^2.

Accepted Answer

The integrating factor is e^(integral of dx/x)=x, so integrating y.x=integral of x^3 dx=x^4/4+C. Answer: y=x^3/4+C/x.

Question 5

How many arbitrary constants does the particular solution of a third-order differential equation contain?

Accepted Answer

A particular solution fixes every arbitrary constant to a specific value using given conditions, regardless of the equation's order. Answer: 0 (zero) arbitrary constants.

Question 6

Find a particular solution of x^2 dy+(xy+y^2)dx=0, given y=1 when x=1.

Accepted Answer

With y=vx, the equation reduces to a separable equation in v and x; integrating and substituting v=y/x, then applying y(1)=1, fixes the constant. Answer: y=2x/(3x^2-1).

Question 7

Find a particular solution of dy/dx+2y tan x=sin x, given y=0 when x=pi/3.

Accepted Answer

The integrating factor is sec^2x; integrating y.sec^2x=integral of sinx.sec^2x dx=secx.tanx-related result gives y=cosx+C.cos^2x, and applying y(pi/3)=0 gives C=-2. Answer: y=cos x-2cos^2 x.

Question 8

In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank. How much will it be worth after 10 years (e^0.5=1.648)?

Accepted Answer

Since dP/dt=(r/100)P, separating and integrating gives P=P0.e^(rt/100); substituting P0=1000, r=5, t=10 gives P=1000.e^0.5. Answer: Rs 1648.

Practice — Differential Equations

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