91Ó°ÊÓ

Show that each function \(y=f(x)\) is a solution of the accompanying differential equation. $$2 y^{\prime}+3 y=e^{-x}$$ a. \(y=e^{-x}\) b. \(y=e^{-x}+e^{-(3 / 2) x}\) c. \(y=e^{-x}+C e^{-(3 / 2) x}\)

Gives a value of \(\sinh\) \(x\) or \(\cosh\) \(x\). Use the definitions and the identity \(\cosh ^{2} x-\sinh ^{2} x=1\) to find the values of the remaining five hyperbolic functions. $$\sinh x=-\frac{3}{4}$$

Evaluate the integrals. $$\int_{-1}^{0} \frac{3 d x}{3 x-2}$$

Gives a value of \(\sinh\) \(x\) or \(\cosh\) \(x\). Use the definitions and the identity \(\cosh ^{2} x-\sinh ^{2} x=1\) to find the values of the remaining five hyperbolic functions. $$\sinh x=\frac{4}{3}$$

Show that each function \(y=f(x)\) is a solution of the accompanying differential equation. $$y^{\prime}=y^{2}$$ a. \(y=-\frac{1}{x}\) b. \(y=-\frac{1}{x+3}\) c. \(y=-\frac{1}{x+C}\)

Evaluate the integrals. $$\int \frac{2 y d y}{y^{2}-25}$$

Show that each function \(y=f(x)\) is a solution of the accompanying differential equation. $$y=\frac{1}{x} \int_{1}^{x} \frac{e^{t}}{t} d t, \quad x^{2} y^{\prime}+x y=e^{x}$$

Gives a value of \(\sinh\) \(x\) or \(\cosh\) \(x\). Use the definitions and the identity \(\cosh ^{2} x-\sinh ^{2} x=1\) to find the values of the remaining five hyperbolic functions. $$\cosh x=\frac{17}{15}, \quad x>0$$

Evaluate the integrals. $$\int \frac{8 r d r}{4 r^{2}-5}$$

Gives a value of \(\sinh\) \(x\) or \(\cosh\) \(x\). Use the definitions and the identity \(\cosh ^{2} x-\sinh ^{2} x=1\) to find the values of the remaining five hyperbolic functions. $$\cosh x=\frac{13}{5}, \quad x>0$$