Chapter 24

Solve the following differential equations: 1\. \(y^{\prime \prime}+9 y=4 e 2^{x}\). Suggestion: Try \(a e^{2 x}\) to obtain the particular integral. Ans. \(y\) \(=A \cos 3 x+B \sin 3 x+\frac{4}{13} e^{2 x}\).

Solve the following differential equations by the method of separation of variables 1\. \(\frac{d y}{d x}=x^{2} y\). Ans. \(-\frac{1}{2 y^{2}}=\frac{x^{3}}{3}+c\)

\(\frac{d y}{d x}=\frac{x^{2}}{y^{3}}\).

(a) A bead is free to slide along a smooth straight horizontal wire. One end of the wire is kept at a fixed point which one can take as the pole or origin and the wire is rotated around the origin in a horizontal plane at a constant angular velocity \(\omega\). Find the equation of the path of the bead. Suggestion: Ignore gravity and use the fact that no radial force acts to keep the bead in one position on the wire. (b) Suppose the bead is at the origin when \(t=0\) and the wire starts rotating at \(\theta=0\). Describe the motion of the bead.

(a) \(y^{\prime \prime}+9 y=\sin 3 x\). Suggestion: To find the particular integral try \(y_{p}=x(a \sin 3 x+\) \(b \cos 3 x\) ). (b) Discuss the relative importance of the complementary function and the particular integral, that is, of the transient and the steady state, for large \(x\).

\(\frac{d y}{d x}=\frac{3 x+x y^{2}}{y+x^{2} y}\).

\(\sin ^{2} y d x+\cos ^{2} x d y=0\) subject to the condition \(y=\pi / 4\) when \(x=\pi / 4\)

\(x \sqrt{1+y^{2}} d x-y \sqrt{1+x^{2}} d y=0\).

\(y^{\prime}=8 x y+3 y\).

Solve \(y^{\prime}=(x+y)^{2}\) by letting \(v=x+y\).