2.1 General first order
2.1.1 Transport equation \(u_t+ u_x = 0\)
2.1.2 Transport equation \(u_t-3 u_x = 0\) IC \(u(0,x)=e^{-x^2}\). Peter Olver textbook, 2.2.2 (a)
2.1.3 Transport equation \(u_t+2 u_x = 0\) IC \(u(-1,x)=\frac {x}{1+x^2}\). Peter Olver textbook, 2.2.2 (b)
2.1.4 Transport equation \(u_t+u_x+\frac {1}{2}u = 0\) IC \(u(0,x)=\arctan (x)\). Peter Olver textbook, 2.2.2 (c)
2.1.5 Transport equation \(u_t-4u_x+u = 0\) IC \(u(0,x)=\frac {1}{1+x^2}\). Peter Olver textbook, 2.2.2 (d)
2.1.6 Transport equation \(u_t+2 u_x= \sin x\) IC \(u(0,x)=\sin x\). Peter Olver textbook, 2.2.5
2.1.7 Transport equation \(u_t+\frac {1}{1+x^2} u_x= 0\) IC \(u(x,0)=\frac {1}{1+(3+x)^2}\). Peter Olver textbook, page 27
2.1.8 Transport equation \(u_t-x u_x= 0\) IC \(u(x,0)=\frac {1}{1+x^2}\). Peter Olver textbook, problem 2.2.17
2.1.9 Transport equation \(u_t+(1-2 t) u_x= 0\) IC \(u(x,0)=\frac {1}{1+x^2}\). Peter Olver textbook, problem 2.2.29
2.1.10 Transport equation \(u_t+\frac {1}{x^2+4} u_x= 0\) IC \(u(x,0)=e^{x^3+12 x}\)
2.1.11 \(3 u_x + 5 u_y = x\)
2.1.12 \(x u_y + y u_x = -4 x y u\) and \(u(x,0)=e^{-x^2}\)
2.1.13 \(u_t + u_x = 0\) and \(u(x,0)=\sin x\) and \(u(0,t)=0\)
2.1.14 \(u_t+ c u_x = 0\) and \(u(x,0)=e^{-x^2}\)
2.1.15 (Haberman 12.2.2) \(\omega _t -3 \omega _x = 0\) and \(\omega (x,0)=\cos x\)
2.1.16 (Haberman 12.2.4) \(\omega _t +c \omega _x = 0\) and \(\omega (x,0)=f(x)\) and \(\omega (0,t)=h(t)\)
2.1.17 (Haberman 12.2.5 (a)) \(\omega _t +c \omega _x = e^{2 x}\) and \(\omega (x,0)=f(x)\)
2.1.18 (Haberman 12.2.5 (d)) \(\omega _t +3 t \omega _x = \omega (x,t)\) and \(\omega (x,0)=f(x)\)
2.1.19 \( 2 u_x + 5 u_y = u^2(x,y) + 1\)
2.1.20 Clairaut equation \(x u_x + y u_y + \frac {1}{2} ( (u_x)^2+ (u_y)^2 ) = 0\)
2.1.21 Clairaut equation. \(x u_x + y u_y + \frac {1}{2} ( (u_x)^2+ (u_y)^2 ) = 0\) with \(u(x,0)= \frac {1}{2} (1-x^2)\)
2.1.22 Clairaut equation. \(u = x u_x+ y u_y + \sin ( u_x + u_y )\)
2.1.23 Recover a function from its gradient vector
2.1.24 \(x f_y - f_x = \frac {g(x)}{h(y)} f^2\)
2.1.25 \(f_x + (f_y)^2 = f(x,y,z)+z\)
2.1.26 \(x u_x+y u_y=u\) (Example 3.5.1 in Lokenath Debnath)
2.1.27 \(x u_x+y u_y=n u\) Example 3.5.2 in Lokenath Debnath
2.1.28 \(x^2 u_x+y^2 u_y=(x+y) u\) Example 3.5.3 in Lokenath Debnath
2.1.29 \((y-z) u_x + (z-x) u_y + (x-y) u_z = 0\) (Example 3.5.4 in Lokenath Debnath)
2.1.30 \(u(x+y) u_x+u(x-y) u_y=x^2+y^2\) (Example 3.5.5 in Lokenath Debnath)
2.1.31 \(u_x-u_y=1\) with \(u(x,0)=x^2\) Example 3.5.6 in Lokenath Debnath
2.1.32 \(y u_x+x u_y=u\) with \(u(x,0)=x^3\) and \(u(0,y)=y^3\) Example 3.5.8 in Lokenath Debnath
2.1.33 \(x u_x+y u_y=x e^{-u}\) with \(u=0\) on \(y=x^2\) Example 3.5.10 in Lokenath Debnath
2.1.34 \(u_t+u u_x=x\) with \(u(x,0)=f(x)\) Example 3.5.11 in Lokenath Debnath.
2.1.35 \(u_x=0\) Problem 3.3(a) Lokenath Debnath
2.1.36 \(a u_x+b u_y=0\) Problem 3.3(b) Lokenath Debnath
2.1.37 \(u_x+y u_y=0\) Problem 3.3(c) Lokenath Debnath
2.1.38 \((1+x^2) u_x+ u_y=0\) Problem 3.3(d) Lokenath Debnath
2.1.39 \(2 x y u_x+(x^2+y^2)u_y=0\) Problem 3.3(e) Lokenath Debnath
2.1.40 \((y+u) u_x+y u_y=x-y\) Problem 3.3(f) Lokenath Debnath
2.1.41 \(y^2 u_x- x y u_y=x(u-2 y)\) Problem 3.3(g) Lokenath Debnath
2.1.42 \(y u_y - x u_x = 1\) Problem 3.3(h) Lokenath Debnath
2.1.43 \(u_x+2 x y^2 u_y=0\) Problem 3.4 Lokenath Debnath
2.1.44 \(3 u_x+2 u_y=0\) with \(u(x,0)=\sin x\). Problem 3.5(a) Lokenath Debnath
2.1.45 \(y u_x+x u_y=0\) with \(u(0,y)=e^{-y^2}\). Problem 3.5(b) Lokenath Debnath
2.1.46 \(x u_x+y u_y=2 x y\) with \(u=2\) on \(y=x^2\). Problem 3.5(c) Lokenath Debnath
2.1.47 \(u_x+x u_y=0\) with \(u(0,y)=\sin y\). Problem 3.5(d) Lokenath Debnath
2.1.48 \(y u_x+x u_y=x y\) with \(u(0,y)=e^{-y^2},u(x,0)=e^{-x^2}\). Problem 3.5(e) Lokenath Debnath
2.1.49 \(u_x+x u_y=(y-\frac {1}{2}x^2)^2\) with \(u(0,y)=e^{y}\). Problem 3.5(f) Lokenath Debnath
2.1.50 \(x u_x+y u_y=u+1\) with \(u=x^2\) on \(y=x^2\) Problem 3.5(g) Lokenath Debnath
2.1.51 \(u u_x - u u_y= u^2 + (x+y)^2\) with \(u(x,0)=1\) Problem 3.5(h) Lokenath Debnath
2.1.52 \(x u_x+(x+y)u_y=u+1\) with \(u(x,0)=x^2\) Problem 3.5(i) Lokenath Debnath
2.1.53 \(x u_x+y u_y+z u_z=0\) Problem 3.8(a) .Lokenath Debnath
2.1.54 \(x^2 u_x+y^2 u_y+z(x+y)u_z=0\) Problem 3.8(b) Lokenath Debnath
2.1.55 \(x(y-z)u_x+y(z-x)u_y+z(x-y)u_z=0\) Problem 3.8(c) Lokenath Debnath
2.1.56 \(y z u_x - x z u_y+ x y (x^2+y^2) u_z=0\) Problem 3.8(d) Lokenath Debnath
2.1.57 \(x(y^2-z^2) u_x + y(z^2-y^2) u_y+ z (x^2-y^2) u_z=0\) Problem 3.8(e) Lokenath Debnath
2.1.58 \(u_x+x u_y=y\) with \(u(0,y)=y^2\) Problem 3.9(a) Lokenath Debnath
2.1.59 \(u_x+x u_y=y\) with \(u(1,y)=2 y\) Problem 3.9(b) Lokenath Debnath
2.1.60 \((u_x+u_y)^2-u^2=0\). Problem 3.10 Lokenath Debnath
2.1.61 \((y+u)u_x+y u_y=x-y\) with \(u(x,1)=1+x\). Problem 3.11 Lokenath Debnath
2.1.62 \(2 x u_x+(x+1) u_y=y\) with \(u(1,y)=2 y\). Problem 3.14(d) Lokenath Debnath
2.1.63 \(x u_x+y u_y=x^2+y^2\) with \(u(x,1)=x^2\). Problem 3.14(e) Lokenath Debnath
2.1.64 \(y^2 u_x+(x y) u_y=x\) with \(u(x,1)=x^2\). Problem 3.14(f) Lokenath Debnath
2.1.65 \(x u_x+y u_y=x y\) with \(u=\frac {x^2}{2}\) at \(y=x\). Problem 3.14(g) Lokenath Debnath
2.1.66 \(u_x+u u_y=1\) with \(u(0,y)=a y\). Problem 3.16(a) Lokenath Debnath
2.1.67 \((y+u)u_x+(x+u)u_y=x+y\). Problem 3.17(a) Lokenath Debnath
2.1.68 \(x u(u^2+x y)u_x - y u(u^2+x y) u_y = x^4\). Problem 3.17(b) Lokenath Debnath
2.1.69 \((x+y) u_x + (x-y)u_y =0\). Problem 3.17(c) Lokenath Debnath
2.1.70 \(y u_x - x u_y = e^u\) with \(u(0,y)=y^2-1\)
2.1.71 \(y u_x - x u_y = e^u\)
2.1.72 \(u_t + x u_x = 0\) with \(u(x,0)=x^2\). Math 5587
2.1.73 \(u_t + t u_x = 0\) with \(u(x,0)=e^x\)
2.1.74 \(2 u_x + 3 u_y = 1\)
2.1.75 \(x u_t - t u_x = 0\)
2.1.76 \(u_t + u_x = 0\) with \(u(x,1)=\frac {x}{1+x^2}\)
2.1.77 \(u_x u_y = 1\)
2.1.78 \(u_x u_y = u\) with \(u(x,0)=0,u(0,y)=0\)
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