91Ó°ÊÓ

Solve the equation given in Problem 2.

\({\left( {{\bf{y - 4x - 1}}} \right)^{\bf{2}}}{\bf{dx - dy = 0}}\)

$\mathbf{x}≡\mathbf{0}$In problems 33-40, Solve the equation given in Problem 2.

${\left(\mathbf{y}\mathbf{-}\mathbf{4}\mathbf{x}\mathbf{-}\mathbf{1}\right)}^{\mathbf{2}}\mathbf{dx}\mathbf{-}\mathbf{dy}\mathbf{=}\mathbf{0}$

Use the method described in Problem 32 to show that the orthogonal trajectories to the family of curves ${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{=}\mathbf{kx}$, ka parameter, satisfy $\mathbf{(}\mathbf{2}\mathbf{y}{\mathbf{x}}^{\mathbf{-}\mathbf{1}}\mathbf{)}\mathbf{dx}\mathbf{+}\mathbf{(}{\mathbf{y}}^{\mathbf{2}}{\mathbf{x}}^{\mathbf{-}\mathbf{2}}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{dy}\mathbf{=}\mathbf{0}$. Find the orthogonal trajectories by solving the above equation. Sketch the family of curves, along with their orthogonal trajectories. [Hint:Try multiplying the equation by ${\mathbf{x}}^{\mathbf{m}}{\mathbf{y}}^{\mathbf{n}}$as in Problem 30.]

Question: In Problems 31-40, solve the initial value problem.

$\frac{\mathrm{dy}}{\mathrm{dx}}\mathbf{-}\frac{2y}{x}\mathbf{=}{\mathbf{x}}^2\mathbf{cosx}\mathbf{,}\text{ â¶Ä‰â¶Ä‰â¶Ä‰}\mathbf{y}\left(\mathrm{Ï€}\right)\mathbf{=}\mathbf{2}$

Using condition (5), show that the right-hand side of (10) is independent of x by showing that its partial derivative with respect to x is zero. [Hint: Since the partial derivatives of M are continuous, Leibniz’s theorem allows you to interchange the operations of integration and differentiation.]

Solve the equation given in Problem 3.

\(\frac{{{\bf{dy}}}}{{{\bf{dx}}}}{\bf{ + }}\frac{{\bf{y}}}{{\bf{x}}}{\bf{ = }}{{\bf{x}}^{\bf{3}}}{{\bf{y}}^{\bf{2}}}\)

Question: In Problems 31-40, solve the initial value problem.

$\left(\mathbf{2}{\mathbf{y}}^2\mathbf{+}\mathbf{4}{\mathbf{x}}^2\right)\mathbf{dx}\mathbf{-}\mathbf{xydy}\mathbf{=}\mathbf{0}\mathbf{,}\text{ â¶Ä‰}\mathbf{y}\left(1\right)\mathbf{=}\mathbf{-}\mathbf{2}$

A pot of boiling water at 100°C is removed from a stove and covered at time t = 0 and left to cool in the kitchen. After 5 min, the water temperature has decreased to 80°C, and another 5 min later it has dropped to 65°C. Assuming Newton’s law of cooling applies, determine the (constant) temperature of the kitchen.

Verify that $\mathbf{F} \mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{}$as defined by (9) and (10) satisfies conditions (4).

In Problems 31-40, solve the initial value problem.

[2 cos(2x+y)-x²] dx+[cos(2x+y) +e^y] dy=0 , y(1)=0