Differential Equations with the Math Sorcerer

Identify separable differential equations by the form dy/dx = f(x) g(y) and separate variables. Integrate both sides to obtain ln|y| = g(x) + C, then y = K e^{g(x)}.

Solve a separable differential equation by partial fractions and natural logs, arriving at P = C e^T / (1 + C e^T).

Learn to solve a separable differential equation by separating variables, multiplying to arrange terms, and integrating to obtain y^4 = 4 ln|x| + C, with constant renaming.

Learn to solve a separable differential equation by separating variables and integrating both sides. Apply the cosine squared identity and the arctangent result to obtain the solution.

Explore a separable differential equation with an initial value by separating variables, integrating, exponentiating, and applying the initial condition x = -1, y = 0 to solve for y.

Explore the theory of exact differential equations, including the total differential, partial derivatives, and the exactness condition. Solve examples by integrating with respect to x and y to find F(x,y).

Identify M and N, compare their partial derivatives dM/dy and dN/dx, and in this example find they are not equal, so the differential equation is not exact.

Solve exact differential equations quickly by checking exactness with dM/dY and dN/dX, then apply the quick method of integrating with respect to x and y and matching terms.

Discover how homogeneous differential equations are defined by degree and solved with substitutions y = u x or x = v y, with a separable example.

solve the Bernoulli differential equation by substituting u = 1/y to transform it into a linear equation, apply the integrating factor, and obtain e^{x/y} = C x.

Solve a Bernoulli differential equation with n=3 by the substitution u = y^(1-n), convert to a linear equation, apply the integrating factor e^{2x}, and express y from u.

Solve first-order differential equations dy/dx = f(ax + by + c) by letting u = ax + by + c, separating, integrating to arctan, and substituting back to obtain y.

Determine whether the functions 9+x, x, and x^2 are linearly independent by forming a linear combination equal to zero, equating coefficients, and concluding all constants must be zero.

Use the characteristic equation to solve a homogeneous linear differential equation with constant coefficients, obtaining complex roots and the solution y = C1 e^{-x/3} cos((√2/3)x) + C2 e^{-x/3} sin((√2/3)x).

Solve a second-order homogeneous linear differential equation by forming the characteristic equation, yielding distinct real roots m1=0 and m2=-1/8, giving y=C1 e^{m1 x}+C2 e^{m2 x} (i.e., y=C1+C2 e^{-x/8}).

solve a homogeneous linear differential equation with constant coefficients using the characteristic equation and the rational roots theorem; identify roots -1 and 5 (multiplicity two) to derive the general solution.

Solve a third-order linear differential equation with constant coefficients by solving the characteristic equation; find roots 0, 8, and -1, yielding Y = C1 + C2 e^{8x} + C3 e^{-x}.

Solve a homogeneous linear differential equation with constant coefficients by the characteristic equation, revealing roots m=0 (multiplicity 2) and m=-1/2±i√3/2, yielding y=C1+C2x+e^{-x/2}[C3 cos(√3/2 x)+C4 sin(√3/2 x)].

Solve a fifth-order linear homogeneous differential equation with constant coefficients by forming the characteristic equation, identify roots m=1 with multiplicity 2, m=-1 with multiplicity 2, and m=-6 with multiplicity 1.

Derive the differential equation from the general solution with distinct real roots 1 and 8, using factors (M-1)(M-8) to obtain the characteristic equation r^2-9r+8=0, yielding y''-9y'+8y=0.

Identify the differential equation from the general solution with alpha=0 and beta=6, yielding roots 0±6i and the characteristic equation m^2+36=0, hence y''+36y=0.

Find the homogeneous linear differential equation with general solution y = C1 e^{-4x} + C2 e^{-3x} by using (m+4)(m+3)=0 to get y''+7y'+12y=0.