Exercises

01

Tricky signs

Check and explain all absolute values and signs (including $C>0$) in the previous example.

02

Separable equations: basics

Solve the equations:

• (a) $x^5{y}^{\prime }+y^5=0$
• (b) $x{y}^2-y^{\prime }{x}^2=0$

03

Separable equations: tricky trig

Solve the equation:

$$x{y}^{\prime }=(1-4x^2)\tan (y).$$

After finding the basic solution, check over the signs and absolute values of everything.

04

Particular solution

Find the particular solution to

$$\frac{y^{\prime }}{1+x^2}=\frac{x}{y}$$

which includes the point $(1,3)$.

05

Linear equations: basics

Solve the following ODEs, giving a complete family of solutions:

• (a) $y^{\prime }+xy=x$,
• (b) $y^{\prime }+y\cot (x)=2x\csc (x)$.

06

Linear equation: particular solution

Find a particular solution of the following ODE which passes through the point $(0,0)$:

$$y^{\prime }-2xy=6x{e}^{x^2}.$$

07

Checking exactness

Which of the following ODEs are exact?

• (a) $(\sin x+\tan y+1)dx+\cos x{\sec }^{2}ydy=0$
• (b) $(y-x^3)dx+(x+y^3)dy=0$
• (c) $(2x{y}^4+\sin y)dx+(4x^2{y}^3+x\cos y)dy=0$
• (d) $3x^2(1+\ln y)dx+\left(\frac{x^3}{y}-2y\right)dy=0$

08

Solving an exact equation

Show that the ODE is exact, and solve it:

$$\left(x+\frac{2}{y}\right)dy+ydx=0.$$