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Magazine Problem 11
Verify that the following functions are solutions to the given differential equation. $$ y=\frac{1}{1-x} \text { solves } y^{\prime}=y^{2} $$
Problem 12
Verify that the following functions are solutions to the given differential equation. $$ y=e^{x^{2} / 2} \text { solves } y^{\prime}=x y $$
Problem 13
Verify that the following functions are solutions to the given differential equation. $$ y=4+\ln x \text { solves } x y^{\prime}=1 $$
Problem 14
Verify that the following functions are solutions to the given differential equation. $$ y=3-x+x \ln x \text { solves } y^{\prime}=\ln x $$
Problem 15
Verify that the following functions are solutions to the given differential equation. $$ y=2 e^{x}-x-1 \text { solves } y^{\prime}=y+x $$
Problem 16
Verify that the following functions are solutions to the given differential equation. $$ y=e^{x}+\frac{\sin x}{2}-\frac{\cos x}{2} \text { solves } y^{\prime}=\cos x+y $$
Problem 17
Verify that the following functions are solutions to the given differential equation. $$ y=\pi e^{-\cos x} \text { solves } y^{\prime}=y \sin x $$
Problem 18
Verify the following general solutions and find the particular solution. Find the particular solution to the differential equation \(y^{\prime}=4 x^{2}\) that passes through \((-3,-30),\) given that \(y=C+\frac{4 x^{3}}{3}\) is a general solution.
Problem 19
Verify the following general solutions and find the particular solution. Find the particular solution to the differential equation \(y^{\prime}=3 x^{3}\) that passes through \((1,4.75),\) given that \(y=C+\frac{3 x^{4}}{4}\) is a general solution.
Problem 20
Verify the following general solutions and find the particular solution. Find the particular solution to the differential equation \(y^{\prime}=3 x^{2} y\) that passes through \((0,12),\) given that \(y=C e^{x^{3}}\) is a general solution.
