MATH383: A first course in differential equationsJanuary 23, 2019

Due date: Jan 30, 2019, 11:55PM.

Bibliography: Trench Chap. 2

1. Exercises 1-4, p. 61

   Ex.1

   $y^{\text{'}}=f(x,y)=(x^2+y^2)/\sin x$. Both $f$ and $f_y=2y/\sin x$ are continous in $y$ except when $x_0=k\pi$. A unique solution is found for some interbal for any $x_0\ne k\pi$, $k\in \mathbb{Z}$, any $y_0$.

   Ex.2

   $y^{\text{'}}=f(x,y)=(e^x+y)/(x^2+y^2)$. Calculate

   ${\displaystyle f_y=\frac{x^2+y^2-2(e^x+y)y}{{(x^2+y^2)}^2}.}$

   A unique solution is found for $(x_0,y_0)\ne (0,0)$.

   Ex.3

   $y^{\text{'}}=f(x,y)=\tan (xy)$. Compute

   ${\displaystyle f_y=\frac{x}{{\cos }^2(xy)}}$

   A unique solution is found for $\cos (x_0{y}_0)\ne 0\Rightarrow x_0{y}_0\ne k\pi +\pi /2$, $k\in \mathbb{Z}$.

   Ex. 4

   $y^{\text{'}}=f(x,y)=(x^2+y^2)/\ln (xy)$. Compute

   ${\displaystyle f_y=\frac{2y\ln (xy)-\frac{x^2+y^2}{y}}{{\ln }^2(xy)}=\frac{2y^2\ln (xy)-x^2-y^2}{y{\ln }^2(xy)}.}$

   A unique solution is found for $x_0{y}_0\ne 1$, $x_0{y}_0>0$.

2. Exercises 15,16, p.61

   Ex.15

   1. For $\left|x\right|>1$, $y=0$ implies $y^{\text{'}}=0$ and hence verifies $y^{\text{'}}=10x{y}^{2/5}/3$. For $\left|x\right|<1$, $y={(x^2-1)}^{5/3}$ implies

      ${\displaystyle y^{\text{'}}=\frac{10x}{3}{(x^2-1)}^{2/3},}$

      and replacing in DE gives

      ${\displaystyle \frac{10x}{3}{(x^2-1)}^{2/3}=\frac{10}{3}x{\left[{(x^2-1)}^{5/3}\right]}^{2/5}=\frac{10}{3}x{(x^2-1)}^{2/3},\mathrm{verified}.}$

      At $x=\pm 1$, $y(\pm 1)=0$, and980

3. Exercises 17,18, p.61

4. Exercises 1-4, p. 79