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Magazine Problem 1
Determining Whether a Differential Equation Is Linear In Exercises \(1-4,\) determine whether the differential equation is linear. Explain your reasoning. $$ x^{3} y^{\prime}+x y=e^{x}+1 $$
Problem 1
Verifying a Solution In Exercises \(1-8,\) verify the solution of the differential equation. $$ y=C e^{4 x} \quad y^{\prime}=4 y $$
Problem 1
Finding a General Solution Using Separation of Variables In Exercises \(1-14,\) find the general solution of the differential equation. $$ \frac{d y}{d x}=\frac{x}{y} $$
Problem 1
Solving a Differential Equation In Exercises \(1-10\) , solve the differential equation. $$ \frac{d y}{d x}=x+3 $$
Problem 2
Determining Whether a Differential Equation Is Linear In Exercises \(1-4,\) determine whether the differential equation is linear. Explain your reasoning. $$ 2 x y-y^{\prime} \ln x=y $$
Problem 2
Solving a Differential Equation In Exercises \(1-10\) , solve the differential equation. $$ \frac{d y}{d x}=5-8 x $$
Problem 2
Finding a General Solution Using Separation of Variables In Exercises \(1-14,\) find the general solution of the differential equation. $$ \frac{d y}{d x}=\frac{3 x^{2}}{y^{2}} $$
Problem 3
Finding a General Solution Using Separation of Variables In Exercises \(1-14,\) find the general solution of the differential equation. $$ x^{2}+5 y \frac{d y}{d x}=0 $$
Problem 3
Verifying a Solution In Exercises \(1-8,\) verify the solution of the differential equation. $$ x^{2}+y^{2}=C y \quad y^{\prime}=\frac{2 x y}{x^{2}-y^{2}} $$
Problem 3
Solving a Differential Equation In Exercises \(1-10\) , solve the differential equation. $$ \frac{d y}{d x}=y+3 $$
