91Ó°ÊÓ

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

Find a function \(y=f(x)\) satisfying the given differential equation and the prescribed initial condition. \(\frac{d y}{d x}=\sqrt{x} ; y(4)=0\)

$$ y^{\prime}+3 y=2 x e^{-3 x} $$

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. $$ \frac{d y}{d x}=y-\sin x $$

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

Verify by substitution that each given function is a solution of the given differential equation. Throughout these problems, primes denote derivatives with respect to \(x\). $$ y^{\prime \prime}=9 y ; y_{1}=e^{3 x}, y_{2}=e^{-3 x} $$

Find a function \(y=f(x)\) satisfying the given differential equation and the prescribed initial condition. \(\frac{d y}{d x}=\frac{1}{x^{2}} ; y(1)=5\)

$$ y^{\prime}-2 x y=e^{x^{2}} $$

Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems 1 through \(18 .\) Primes denote derivatives with respect to \(x.\) $$ (1+x) \frac{d y}{d x}=4 y $$

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