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Magazine Problem 6
Finding a General Solution Using Separation of Variables In Exercises 5-18, find the general solution of the differential equation. $$\frac{d y}{d x}=\frac{3 x^{2}}{y^{2}}$$
Problem 6
In Exercises \(3-6,\) determine whether the differential equation is linear. Explain your reasoning. $$\frac{2-y^{\prime}}{y}=5 x$$
Problem 6
Verifying a Solution In Exercises \(5-10\) , verify that the function is a solution of the differential equation. $$\begin{array}{ll}{\text { Function }} & {\text { Differential Equation }} \\\ {y=e^{-2 x}} & {3 y^{\prime}+5 y=-e^{-2 x}}\end{array}$$
Problem 7
Verifying a Solution In Exercises \(5-10\) , verify that the function is a solution of the differential equation. $$\begin{array}{ll}{\text { Function }} & {\text { Differential Equation }} \\\ {y=C_{1} \sin x-C_{2} \cos x} & {y^{\prime \prime}+y=0}\end{array}$$
Problem 7
Finding a General Solution Using Separation of Variables In Exercises 5-18, find the general solution of the differential equation. $$\frac{d y}{d x}=\frac{x-1}{y^{3}}$$
Problem 7
Solving a Differential Equation In Exercises \(3-12\) , find the general solution of the differential equation. $$y^{\prime}=\frac{5 x}{y}$$
Problem 7
In Exercises \(7-14,\) find the general solution of the first-order linear differential equation for \(x>0 .\) $$\frac{d y}{d x}+\left(\frac{1}{x}\right) y=6 x+2$$
Problem 8
Solving a Differential Equation In Exercises \(3-12\) , find the general solution of the differential equation. $$y^{\prime}=-\frac{\sqrt{x}}{4 y}$$
Problem 8
Finding a General Solution Using Separation of Variables In Exercises 5-18, find the general solution of the differential equation. $$\frac{d y}{d x}=\frac{6-x^{2}}{2 y^{3}}$$
Problem 8
In Exercises \(7-14,\) find the general solution of the first-order linear differential equation for \(x>0 .\) $$\frac{d y}{d x}+\left(\frac{2}{x}\right) y=3 x-5$$
