91Ó°ÊÓ

Solve the following differential equations. . \(\left(D^{2}-2 D+1\right) y=0\)

Find the general solution of each of the following differential equations. \(y^{\prime} \cos x+y=\cos ^{2} x\)

Solve the following differential equations. . \((x-y) d y+(y+x+1) d x=0\)

Identify each of the differential equations as to type (for example, separable, linear first order, linear second order, etc.), and then solve it. \((2 x-y \sin 2 x) d x+\left(2 y-\sin ^{2} x\right) d y=0\)

Solve the following differential equations. \(\left(D^{2}+16\right) y=0\)

Find the general solution of each of the following differential equations. \(y^{\prime} \sqrt{x^{2}+1}+x y=x\)

The curvature of a curve in the \((x, y)\) plane is $$ K=y^{n} /\left(1+y^{\prime 2}\right)^{3 / 2} $$ With \(K=\) const., solve this differential equation to show that curves of constant curvature are circles (or straight lines).

Identify each of the differential equations as to type (for example, separable, linear first order, linear second order, etc.), and then solve it. \(y^{\prime \prime}+2 y^{\prime}+2 y=10 e^{x}+6 e^{-x} \cos x\)

Find the "general solution" (that is, a solution containing an arbitrary constant) of each of the following differential equations, by separation of variables. Then find a particular solution of each equation satisfying the given boundary conditions. \(y^{\prime}=\frac{2 x y^{2}+x}{x^{2} y-y}\) \(y=0\) when \(x=\sqrt{2}\)

Solve the following differential equations. \(\left(D^{2}-5 D+6\right) y=0\)