91Ó°ÊÓ

$$ x y^{\prime}+2 y=3 x, y(1)=5 $$

Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems 1 through \(18 .\) Primes denote derivatives with respect to \(x.\) $$ 2 \sqrt{x} \frac{d y}{d x}=\sqrt{1-y^{2}} $$

We have provided the slope field of the indicated differential equation, together with one or more solution curves. Sketch likely solution curves through the additional points marked in each slope field. $$ \frac{d y}{d x}=y-x+1 $$

Find a function \(y=f(x)\) satisfying the given differential equation and the prescribed initial condition. \(\frac{d y}{d x}=\frac{1}{\sqrt{x+2}} ; y(2)=-1\)

$$ x(x+y) y^{\prime}=y(x-y) $$

Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems 1 through \(18 .\) Primes denote derivatives with respect to \(x.\) $$ \frac{d y}{d x}=3 \sqrt{x y} $$

Verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to \(x\). $$ y^{\prime \prime}+4 y^{\prime}+4 y=0 ; y_{1}=e^{-2 x}, y_{2}=x e^{-2 x} $$

$$ x y^{\prime}+5 y=7 x^{2}, y(2)=5 $$

We have provided the slope field of the indicated differential equation, together with one or more solution curves. Sketch likely solution curves through the additional points marked in each slope field. $$ \frac{d y}{d x}=x-y+1 $$

$$ (x+2 y) y^{\prime}=y $$