91Ó°ÊÓ

Find a function \(y=f(x)\) satisfying the given differential equation and the prescribed initial condition. \(\frac{d y}{d x}=x \sqrt{x^{2}+9} ; y(-4)=0\)

Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems 1 through \(18 .\) Primes denote derivatives with respect to \(x.\) $$ \frac{d y}{d x}=(64 x y)^{1 / 3} $$

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

Find a function \(y=f(x)\) satisfying the given differential equation and the prescribed initial condition. \(\frac{d y}{d x}=\frac{10}{x^{2}+1} ; y(0)=0\)

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

Find a function \(y=f(x)\) satisfying the given differential equation and the prescribed initial condition. \(\frac{d y}{d x}=\cos 2 x ; y(0)=1\)

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

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

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

Find a function \(y=f(x)\) satisfying the given differential equation and the prescribed initial condition. \(\frac{d y}{d x}=\frac{1}{\sqrt{1-x^{2}}} ; y(0)=0\)