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Magazine Problem 4
Show that each function \(y=f(x)\) is a solution of the accompanying differential equation. $$y=\frac{1}{\sqrt{1+x^{4}}} \int_{1}^{x} \sqrt{1+t^{4}} d t, \quad y^{\prime}+\frac{2 x^{3}}{1+x^{4}} y=1$$
Problem 5
Rewrite the expressions in terms of exponentials and simplify the results as much as you can. $$2 \cosh (\ln x)$$
Problem 5
Show that each function is a solution of the given initial value problem. $$\begin{array}{lll} \text { Differential } & \text { Initial } & \text { Solution } \\ \text { equation } & \text { equation } & \text { candidate } \\\ \hline y^{\prime}+y=\frac{2}{1+4 e^{2 x}} & y(-\ln 2)=\frac{\pi}{2} & y=e^{-x} \tan ^{-1}\left(2 e^{x}\right)\end{array}$$
Problem 5
Evaluate the integrals. $$\int \frac{3 \sec ^{2} t}{6+3 \tan t} d t$$
Problem 6
Evaluate the integrals. $$\int \frac{\sec y \tan y}{2+\sec y} d y$$
Problem 6
Rewrite the expressions in terms of exponentials and simplify the results as much as you can. $$\sinh (2 \ln x)$$
Problem 6
Show that each function is a solution of the given initial value problem. $$\begin{array}{lll} \text { Differential } & \text { Initial } & \text { Solution } \\ \text { equation } & \text { equation } & \text { candidate } \\\ \hline y^{\prime}=e^{-x^{2}}-2 x y & y(2)=0 & y=(x-2) e^{-x^{2}}\end{array}$$
Problem 7
Evaluate the integrals. $$\int \frac{d x}{2 \sqrt{x}+2 x}$$
Problem 7
Rewrite the expressions in terms of exponentials and simplify the results as much as you can. $$\cosh 5 x+\sinh 5 x$$
Problem 7
Show that each function is a solution of the given initial value problem. $$\begin{array}{lll} \text { Differential } & \text { Initial } & \text { Solution } \\ \text { equation } & \text { equation } & \text { candidate } \\\ \hline x y^{\prime}+y=-\sin x, x>0 & y\left(\frac{\pi}{2}\right)=0 & y=\frac{\cos x}{x}\end{array}$$
