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Magazine Problem 1
State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). $$(1-x) y^{\prime \prime}-4 x y^{\prime}+5 y=\cos x$$
Problem 2
State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). $$x \frac{d^{3} y}{d x^{3}}-\left(\frac{d y}{d x}\right)^{4}+y=0$$
Problem 3
\(y=1 /\left(x^{2}+c\right)\) is a one-parameter family of solutions of the first-order DE \(y^{\prime}+2 x y^{2}=0 .\) Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval \(I\) over which the solution is defined. $$y(2)=\frac{1}{3}$$
Problem 4
State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). $$\frac{d^{2} u}{d r^{2}}+\frac{d u}{d r}+u=\cos (r+u)$$
Problem 4
\(y=1 /\left(x^{2}+c\right)\) is a one-parameter family of solutions of the first-order DE \(y^{\prime}+2 x y^{2}=0 .\) Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval \(I\) over which the solution is defined. $$y(-2)=\frac{1}{2}$$
Problem 5
\(y=1 /\left(x^{2}+c\right)\) is a one-parameter family of solutions of the first-order DE \(y^{\prime}+2 x y^{2}=0 .\) Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval \(I\) over which the solution is defined. $$y(0)=1$$
Problem 5
State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). $$\frac{d^{2} y}{d x^{2}}=\sqrt{1+\left(\frac{d y}{d x}\right)^{2}}$$
Problem 6
State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). $$\frac{d^{2} R}{d t^{2}}=-\frac{k}{R^{2}}$$
Problem 6
\(y=1 /\left(x^{2}+c\right)\) is a one-parameter family of solutions of the first-order DE \(y^{\prime}+2 x y^{2}=0 .\) Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval \(I\) over which the solution is defined. $$y\left(\frac{1}{2}\right)=-4$$
Problem 7
State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). $$(\sin \theta) y^{\prime \prime \prime}-(\cos \theta) y^{\prime}=2$$
