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.

Solve an initial value problem for a separable differential equation by integrating to get y^3 = -4 cosine x + 12, then apply y(0)=2 and take the cube root.

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.

Solve a first order linear differential equation using an integrating factor. Multiply by secant, integrate, and obtain y = sin x + C cos x with attention to domain.

Differential equations with the Math Sorcerer: solve a first order linear differential equation by rewriting in standard form, applying the integrating factor, and integrating to obtain the solution.

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.

Determine if the differential equation is exact by computing M_y and N_x; the example shows using product and chain rules, concluding the equation is not exact.

Confirm exactness by comparing ∂M/∂y and ∂N/∂x, then construct F with F_x=M and F_y=N. The result is F(x,y)=cos x sin y - ln|cos x| = C.

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.

solves a homogeneous differential equation using y = u x substitution, reveals separability, and yields the solution y = x (C − ln|x|).

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.

Explore two functions, x squared and x times the absolute value of x, proving they are linearly independent while their Wronskian is zero, with a piecewise graph and determinant calculation.

Solve homogeneous linear differential equations with constant coefficients by forming the auxiliary equation, evaluating distinct real, repeated real, and complex conjugate roots, and deriving the corresponding general solutions.

Solve y'' + 36y = 0 via the characteristic equation r^2 + 36 = 0, yielding roots ±6i and general solution y = C1 cos 6x + C2 sin 6x.

Solve a second-order homogeneous differential equation by forming the auxiliary equation m^2 - 36 = 0, yielding distinct roots m = ±6 and general solution C1 e^{6x} + C2 e^{-6x}.

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.

From the general solution with roots 0 and 3, form the characteristic equation m(m-3)=0. This yields the differential equation y'' - 3y' = 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.