Hard
Elimination Methods
Solve the system of differential equations:
$$
\frac{dy}{dx} = y + z + x,\qquad \frac{dz}{dx} = -4y - 3z + 2x
$$
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Hard
Elimination Methods
Solve the system of differential equations:
$$
\frac{dx}{dt} = x + 2y + t,\qquad \frac{dy}{dt} = 2x + y + t
$$
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Hard
Elimination Methods
Solve the system of differential equations:
$$
\dot x = y - 2e^{t},\qquad \dot y = x + t^2
$$
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Hard
Elimination Methods
Solve the system of differential equations:
$$
\dot x = y - 5\cos t,\qquad \dot y = 2x + y
$$
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Easy
Elimination Methods
Solve the system of differential equations:
$$
\frac{dx}{dt} = y + z,\qquad \frac{dy}{dt} = x + z,\qquad \frac{dz}{dt} = x + y
$$
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Medium
Elimination Methods
Solve the system of differential equations:
$$
\frac{dy}{dx} = 1 - \frac{1}{z},\qquad \frac{dz}{dx} = \frac{1}{y-x}
$$
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Easy
Elimination Methods
Solve the system of differential equations:
$$
\frac{d^2y}{dx^2} = z,\qquad \frac{d^2z}{dx^2} = y
$$
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Medium
Elimination Methods
Solve the system of differential equations:
$$
\ddot x = 2y,\qquad \ddot y = -2x
$$
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Medium
Elimination Methods
Solve the system of differential equations:
$$
\ddot x + x = \dot y,\qquad 4\dot x + 2x = \dot y + 2y,\quad x(0)=0,\ y(0)=1,\ \dot x(0)=2
$$
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Medium
Elimination Methods
Solve the system of differential equations:
$$
\ddot x + \dot y + x = e^t,\qquad \dot x + \ddot y = 1
$$
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Hard
Elimination Methods
Solve the system of differential equations:
$$
2\ddot x+2\dot x+x+3\ddot y+\dot y+y=0,\qquad \ddot x+4\dot x-x+3\ddot y+2\dot y-y=0
$$
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Hard
Elimination Methods
Solve the system of differential equations:
$$
6\ddot x-3\dot x+6x+\dot y=t,\qquad \dot x+8x+y=0,\quad x(0)=\tfrac{11}{36},\ y(0)=-\tfrac{22}{9},\ \dot x(0)=\tfrac16
$$
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Medium
Elimination Methods
Solve the system of differential equations:
$$
\ddot x = 3x + 4\dot y,\qquad \ddot y = 3y + 4\dot x,\quad x(0)=2,\ y(0)=2,\ \dot x(0)=0,\ \dot y(0)=0
$$
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Hard
Elimination Methods
Solve the system of differential equations:
$$
5\ddot x+5x+3\ddot y+3y+8\ddot z+8z=0,\quad \ddot x+4x+3\ddot y+12y+3\ddot z+12z=0,\quad \ddot x-2x-2\ddot y-11y-9z=0
$$
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Easy
Elimination Methods
Solve the system of differential equations:
$$
\frac{dx}{1} = \frac{dy}{y+z} = \frac{dz}{x+y+z}
$$
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Easy
Elimination Methods
Solve the system of differential equations:
$$
\frac{dx}{dt} = y,\qquad \frac{dy}{dt} = x
$$
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Easy
Elimination Methods
Solve the system of differential equations:
$$
\frac{dx}{dt} = \frac{y}{y-x},\qquad \frac{dy}{dt} = \frac{x}{y-x}
$$
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Medium
Elimination Methods
Solve the system of differential equations:
$$
\dot x = x(y^2-z^2),\qquad \dot y = -y(x^2+z^2),\qquad \dot z = z(x^2+y^2)
$$
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Easy
Elimination Methods
Solve the system of differential equations:
$$
\dot x = y - z,\qquad \dot y = x^2 + y,\qquad \dot z = x^2 + z
$$
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Medium
Elimination Methods
Solve the system of differential equations:
$$
\frac{dx}{y-z} = \frac{dy}{z-x} = \frac{dz}{y-x}
$$
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Easy
Elimination Methods
Solve by finding first integrals the symmetric system:
$$
\frac{dx}{xz} = \frac{dy}{yz} = \frac{dz}{-xy}
$$
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Easy
Elimination Methods
Solve by finding first integrals the symmetric system:
$$
\frac{dx}{y+z} = \frac{dy}{x+z} = \frac{dz}{x+y}
$$
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Medium
Elimination Methods
Solve by finding first integrals the symmetric four-variable system:
$$
\frac{dx}{y-u} = \frac{dy}{z-x} = \frac{dz}{u-y} = \frac{du}{x-z}
$$
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Medium
Elimination Methods
Solve by finding first integrals the symmetric system:
$$
\frac{dx}{x+y^2+z^2} = \frac{dy}{y} = \frac{dz}{z}
$$
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Medium
Elimination Methods
Solve by finding first integrals the symmetric system:
$$
\frac{dx}{x(y+z)} = \frac{dy}{z(z-y)} = \frac{dz}{y(y-z)}
$$
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Easy
Elimination Methods
Solve by finding first integrals the symmetric system:
$$
\frac{dx}{-x^2} = \frac{dy}{xy-2z^2} = \frac{dz}{xz}
$$
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Easy
Elimination Methods
Solve by finding first integrals the symmetric system:
$$
\frac{dx}{x(y^2-z^2)} = \frac{dy}{-y(z^2+x^2)} = \frac{dz}{z(x^2+y^2)}
$$
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Easy
Elimination Methods
Solve by finding first integrals the symmetric system:
$$
\frac{dx}{x} = \frac{dy}{y} = \frac{dz}{z - \sqrt{x^2+y^2+z^2}}
$$
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Medium
Elimination Methods
Solve by finding first integrals the system:
$$
\frac{dx}{dt} = \frac{x-y}{z-t},\qquad \frac{dy}{dt} = \frac{x-y}{z-t},\qquad \frac{dz}{dt} = x-y+1
$$
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Easy
Elimination Methods
Solve by finding first integrals the symmetric system:
$$
\frac{dx}{y(x+y)} = \frac{dy}{-x(x+y)} = \frac{dz}{-(x-y)(2x+2y+z)}
$$
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Hard
Matrix Methods
Solve the symmetric system, subject to the given initial data:
$$
\frac{dx}{x^2-yz} = \frac{dy}{y^2-yz} = \frac{dz}{z(x+y)},\qquad y=1,\ z=-1\ \text{at }x=0
$$
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Medium
Matrix Methods
Verify whether the following candidate functions are first integrals of the symmetric system:
$$
\frac{dx}{x} = \frac{dy}{-y} = \frac{dz}{z};\qquad \text{test } \Phi_1=\frac{x+y}{z+x},\ \ \Phi_2=\frac{z-y}{x+y}
$$
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Medium
Matrix Methods
Substance $A$ (initial amount $a$) decomposes into products $P$ and $Q$, whose amounts $x(t),y(t)$ satisfy:
$$
\frac{dx}{dt} = k_1(a-x-y),\qquad \frac{dy}{dt} = k_2(a-x-y)
$$
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Medium
Matrix Methods
A bacterial population $N(t)$ grows while producing a poison $Q(t)$ that kills it back, according to:
$$
\dot N = N(\alpha-\beta Q),\qquad \dot Q = \gamma N
$$
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Medium
Matrix Methods
A particle moves under a force directed away from the $Oz$ axis, proportional to its distance $\rho=\sqrt{x^2+y^2}$ from that axis, giving Newton's equations:
$$
\ddot x = kx,\qquad \ddot y = ky,\qquad \ddot z = 0
$$
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Hard
Matrix Methods
Two inductively coupled loops (self-inductances $L_1,L_2$, mutual inductance $M$, resistances $R_1,R_2$) are connected to a constant EMF $E$, with zero initial currents. Find $i_1(t)$ and $i_2(t)$:
$$
L_1\frac{di_1}{dt} + M\frac{di_2}{dt} + R_1 i_1 = E,\qquad M\frac{di_1}{dt} + L_2\frac{di_2}{dt} + R_2 i_2 = 0
$$
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Easy
Matrix Methods
Solve the system of differential equations:
$$
\frac{dx_1}{dt} = 2x_1 - x_2,\qquad \frac{dx_2}{dt} = x_1
$$
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Easy
Matrix Methods
Solve the system of differential equations:
$$
\frac{dx_1}{dt} = x_1 - x_2,\qquad \frac{dx_2}{dt} = x_1 + 3x_2
$$
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Easy
Matrix Methods
Solve the system of differential equations:
$$
\frac{dx_1}{dt} = x_1 - 5x_2,\qquad \frac{dx_2}{dt} = 2x_1 - x_2
$$
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Easy
Matrix Methods
Solve the system of differential equations:
$$
\frac{dx_1}{dt} = x_1-x_2+x_3,\qquad \frac{dx_2}{dt} = x_1+x_2-x_3,\qquad \frac{dx_3}{dt} = 2x_1-x_2
$$
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Medium
Matrix Methods
Solve the system of differential equations:
$$
\dot x_1=x_1-x_2-x_3,\qquad \dot x_2=x_1+x_2,\qquad \dot x_3=3x_1+x_3
$$
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Medium
Matrix Methods
Solve the system of differential equations:
$$
\dot x_1=4x_1-x_2,\qquad \dot x_2=3x_1+x_2-x_3,\qquad \dot x_3=x_1+x_3
$$
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Easy
Matrix Methods
Solve the system of differential equations:
$$
\dot x_1=x_2+x_3,\qquad \dot x_2=x_1+x_3,\qquad \dot x_3=x_1+x_2
$$
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Easy
Matrix Methods
Solve the initial value problem:
$$
\dot x=-x+y+z,\qquad \dot y=x-y+z,\qquad \dot z=x+y+z,\qquad x(0)=1,\ y(0)=0,\ z(0)=0
$$
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Medium
Matrix Methods
Solve the system of differential equations:
$$
\dot x=2x+y-2z-t+2,\qquad \dot y=1-x,\qquad \dot z=x+y-z-t+1
$$
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Easy
Matrix Methods
Solve the system of differential equations:
$$
\dot x=3x-4y+e^{-2t},\qquad \dot y=x-2y-3e^{-2t}
$$
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Easy
Matrix Methods
Solve the system of differential equations:
$$
\dot x=x+2y,\qquad \dot y=x-5\sin t
$$
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Easy
Matrix Methods
Solve the system of differential equations:
$$
\dot x=x+y+1+e^t,\qquad \dot y=3x-y
$$
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Easy
Matrix Methods
Solve the system of differential equations:
$$
\dot x=4x-3y+\sin t,\qquad \dot y=2x-y-2\cos t
$$
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Medium
Matrix Methods
Solve the system of differential equations:
$$
\dot x=2x+y+2e^t,\qquad \dot y=x+2y-3e^{4t}
$$
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Easy
Matrix Methods
Solve the system of differential equations:
$$
\dot x=x-y+8t,\qquad \dot y=5x-y
$$
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Easy
Matrix Methods
Solve the system of differential equations:
$$
\dot x=3x-\tfrac12 y-3t^2-\tfrac12 t+\tfrac32,\qquad \dot y=2y-2t-1
$$
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Medium
Matrix Methods
Solve the system of differential equations:
$$
\dot x=-x+2y,\qquad \dot y=-3x+4y+\frac{e^{3t}}{e^{2t}+1}
$$
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Medium
Matrix Methods
Solve the system of differential equations:
$$
\dot x=-4x-2y+\frac{2}{e^t-1},\qquad \dot y=6x+3y-\frac{3}{e^t-1}
$$
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Medium
Matrix Methods
Solve the system of differential equations:
$$
\dot x=x-y+\sec t,\qquad \dot y=2x-y
$$
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Easy
Matrix Methods
Solve the initial value problem:
$$
\dot x=3x+8y,\qquad \dot y=-x-3y,\qquad x(0)=6,\ y(0)=-2
$$
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Medium
Matrix Methods
Solve the system of differential equations:
$$
\dot x=3x-2y,\qquad \dot y=2x-y+15e^{t}\sqrt{t}
$$
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Medium
Matrix Methods
Solve the initial value problem:
$$
\dot x=-4(x+y),\qquad \dot y=-y-\tfrac14\dot x,\qquad x(0)=1,\ y(0)=0
$$
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Medium
Matrix Methods
Solve the initial value problem:
$$
\dot x=x+y+t,\qquad \dot y=x-2y+2t,\qquad x(0)=-\tfrac79,\ y(0)=-\tfrac59
$$
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Hard
Matrix Methods
Solve the initial value problem, in which the derivatives appear mixed together rather than isolated:
$$
\dot x+2\dot y=17x+8y,\qquad 13\dot x-\dot y=53x+2y,\qquad x(0)=2,\ y(0)=1
$$
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Medium
Phase-Plane Analysis
Solve the system of differential equations:
$$
t\dot x-x-3y=t,\qquad t\dot y-x+y=0
$$
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Easy
Phase-Plane Analysis
Solve the system of differential equations:
$$
t\dot x+6x-y=0,\qquad t\dot y+20x-6y=0
$$
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Medium
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=y,\qquad \dot y=z,\qquad \dot z=x
$$
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Medium
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=-x+y,\qquad \dot y=-y+4z,\qquad \dot z=x-4z
$$
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Easy
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=x-2y-z,\qquad \dot y=-x+y+z,\qquad \dot z=x-z
$$
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Easy
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=3x-y+z,\qquad \dot y=x+y+z,\qquad \dot z=4x-y+4z
$$
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Easy
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=-3x+4y-2z,\qquad \dot y=x+z,\qquad \dot z=6x-6y+5z
$$
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Hard
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=2x-y+2z,\qquad \dot y=x+2z,\qquad \dot z=-2x+y-z
$$
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Medium
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=4x-y-z,\qquad \dot y=x+2y-z,\qquad \dot z=x-y+2z
$$
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Medium
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=2x-y-z,\qquad \dot y=3x-2y-3z,\qquad \dot z=-x+y+2z
$$
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Medium
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=-2x+y-2z,\qquad \dot y=x-2y+2z,\qquad \dot z=3x-3y+5z
$$
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Medium
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=3x-2y-z,\qquad \dot y=3x-4y-3z,\qquad \dot z=2x-4y
$$
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Hard
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=x-y+z,\qquad \dot y=x+y-z,\qquad \dot z=-y+2z
$$
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Hard
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=-x+y-2z,\qquad \dot y=4x+y,\qquad \dot z=2x+y-z
$$
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Easy
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=6x-72y+44z,\qquad \dot y=4x-43y+26z,\qquad \dot z=6x-63y+38z
$$
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Easy
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=4y+z,\qquad \dot y=z,\qquad \dot z=4y
$$
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Easy
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=y+z,\qquad \dot y=3x+z,\qquad \dot z=3x+y
$$
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Medium
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=8y,\qquad \dot y=-2z,\qquad \dot z=2x+8y-2z
$$
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Easy
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=-2x-2y-4z,\qquad \dot y=-2x+y-2z,\qquad \dot z=5x+2y+7z
$$
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Hard
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=x+y+z,\qquad \dot y=y+z,\qquad \dot z=x
$$
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Easy
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=2x-y+z,\qquad \dot y=x+z,\qquad \dot z=-3x+y-2z
$$
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Hard
Phase-Plane Analysis
Solve the system of differential equations:
$$
\dot x=y-z-t+3,\qquad \dot y=7-2z,\qquad \dot z=2y-2t
$$
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Medium
Phase-Plane Analysis
Solve the Euler-type (equidimensional) system:
$$
3t\dot x=2x+y-z,\qquad 2t\dot y=x+3y+z,\qquad 6t\dot z=-x+7y+5z
$$
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Hard
Phase-Plane Analysis
Solve the non-homogeneous Euler-type system:
$$
t\dot x=4x+y+2z-\sin t-2\cos t,\qquad t\dot y=-2x+y-2z-\sin t+2\cos t+t\cos t,\qquad t\dot z=-x-y+z+\sin t-\cos t-t\sin t
$$
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