IGNOU MTE 8 2026 - Solve the following DEs(i) .(ii) .b) The differential

Question

Solve the following DEs
(i) $\left( \frac{dy}{dx} - 1 \right)^2 \left( \frac{d^2y}{dx^2} + 1 \right)^2 y = \sin^2(x/2) + e^x + x$ .
(ii) $2x^2y \left( \frac{d^2y}{dx^2} \right) + 4y^2 = x^2 \left( \frac{dy}{dx} \right)^2 + 2xy \left( \frac{dy}{dx} \right)$ .
b) The differential equation of a damped vibrating system under the action of an external periodic force is:

$$\frac{d^2x}{dt^2} + 2m_0 \frac{dx}{dt} + n^2x = a \cos pt$

Show that, if n > m₀ > 0 the complementary function of the differential equation represents vibrations which are soon damped out. Find the particular integral in terms of periodic functions.

09 Jan 2026

Answer :

Word Count : 222

(i) Let (p=\dfrac{dy}{dx}). Then the given equation is ((p-1)^2\left(\dfrac{d^2y}{dx^2}+1\right)^2y=\sin^2\frac{x}{2}+e^x+x). Since (\sin^2\frac{x}{2}=\dfrac{1-\cos x}{2}), the right side is a known function of (x). A particular solution can be sought by assuming (p-1=0) or (\dfrac{d^2y}{dx^2}+1=0). If (p-1=0), then (\dfrac{dy}{dx}=1) giving (y=x+C). Substituting, the left side vanishes, which is not equal to the right side, so this does not satisfy the equation. Hence take (\dfrac{d^2y}{dx^2}+1=0), giving (\dfrac{d^2y}{dx^2}=-1). Integrating,

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