Differential Equations: Learn Basics of Diff. Equation

Learn to form differential equations from a family of curves by eliminating parameters, with one-parameter y = a e^x and y = a sin 2x + b cos 2x.

Form differential equations from given relations, differentiate with respect to x, divide through, and eliminate to obtain the equation of the family.

Learn how to form a second-order differential equation by differentiating a relation with respect to x and handling constants a and b to obtain an equation for y.

Perturbation explains representing curve families by differential equations and eliminating parameters A and B through differentiation. Derivations yield second-order equations, such as 2 y'' + (y')^3 = 0.

Derive a differential equation for circles tangent to the y-axis at the origin by differentiating the circle equation and eliminating the radius.

Derive the differential equation for parabolas with vertex at origin and axis along positive y by differentiating x^2=4ay and eliminating a: x dy/dx - 2y = 0.

Derive a differential equation for the family of circles with centers on the y-axis and radius 3 by differentiating x^2+(y−b)^2=9 to eliminate b, obtaining (x^2−9)(dy/dx)^2 + x^2 = 0.

Differentiate the given solution to obtain the differential equation and verify it matches; illustrated with y'' + y = 0 and y = cos x − sin x.

Assess six candidate functions to verify which satisfy the given differential equations by differentiating and comparing to the defining relation, illustrating a method for checking solutions.

Learn how to verify differential equation solutions by differentiating proposed functions, including linear and exponential forms, and confirm correctness of the given solutions.

Verify that the given functions solve their differential equations: y''+4y=0 with y=3 sin 2x; (1+x^2) y' = x y with y= sqrt(1+x^2); and y'''=0 with y= x^2+2x+1.

Learn how to solve first-order differential equations by identifying general, particular, and singular solutions, and apply separation of variables and substitutions to reduce to separable form.

Explain how to solve basic differential equations using separation of variables and log properties. Derive solutions with constants of integration, including exponential and algebraic forms.

Explore solving differential equations by separation of variables, perform integration, and obtain the solution with the constant of integration.

Learn to solve a separable differential equation by separating variables, integrating, and obtaining an exponential solution with a constant of integration.

Practice solving differential equations by separation of variables and integrating both sides, with examples involving x and y and forms like 1+x^2 and x^2.

Apply separation of variables and integration to solve a differential equation, using logarithm laws and modulus considerations to obtain the final solution.

Solve two initial value problems for a differential equation: the first gives y = x with y(0)=1; the second gives y = x^(5/2) with y(1)=1, x ≠ 0.

Solve two initial value problems by separation of variables; obtain y = 1/(2 - e^(2x)) with y(0)=1, and y = 1/(2x + 2/pi) with y(0)=pi/2.

Solve dy/dx+2y^2=0 by separation for x=1,y=1 to yield y=1/(2x-1); and with cos x cos y=C from the second initial condition, yielding C=1/√2, so y=arccos(1/(√2 cos x)).

Apply separation of variables to solve a differential equation with an initial condition, deriving an explicit y(x) from integration. Explore a second separable example and obtain y = e^{x-1}/x.

Solve differential equations by separation of variables and integration, applying initial conditions. Derive y = x e^{1-1/x} for the first problem and y = -log|x| for the second.

Learn the basic concepts of homogeneous differential equations, where dy/dx is a function of y/x. Solve by substituting y = v x and then applying variable separation.

Demonstrates solving a homogeneous differential equation by checking homogeneity, substituting y = v x, separating variables, and integrating to obtain the implicit solution.

Solve a homogeneous first-order differential equation by substituting y = v x, converting to a separable equation, and obtaining an implicit solution involving y/x and a logarithm.

Explore the basic concepts of linear differential equations in standard form, and solve first-order equations by multiplying by the integrating factor e^(∫P dx), yielding the solution for y.

Solve linear differential equation y' + 2y = e^{3x} by identifying p = 2, applying the integrating factor e^{2x}, and obtaining y = (1/5) e^{3x} + C e^{-2x}.

Solve y' + 3y = e^{mx} with integrating factor e^{3x}, yielding y = e^{mx}/(m+3) + C e^{-3x}; for m ≠ -3, and y = x e^{-3x} + C e^{-3x} for m = -3.

Identify and solve a linear differential equation dy/dx + (1/x) y = x^3 using the integrating factor x. Arrive at y = (1/5) x^4 + C/x, valid for x ≠ 0.

Solve linear differential equations with initial value problems by using given x and y values to determine the solution.

solve a linear first-order differential equation using integrating factor, and apply the initial value condition y(0)=1/2 to find y(x) = (1/2) e^x.

Solve the initial value problem x dy/dx + y = x log x with y(1)=1/4 using an integrating factor. The solution is y = (x/2) log x - (x/4) + 1/(2x).

Convert to standard linear form and apply the integrating factor to solve the initial value problem, yielding y = x - log x - 1 with y(1) = 0.

Apply integrating factor to the linear differential equation y' - y = e^x with the initial value y(0)=1 and derive the solution y = (x+1) e^x.

Differentiating S = 4πr^2 and using dS/dt = k yields r^2 = 8t + 9, so r(t) = sqrt(8t + 9) units.

Solve dp/dt = 0.08 p by separation to get log(p/p0) = (2/25) t. Therefore the doubling time is t = 25/2 log 2 years.

Derive the differential equation from dy/dx = 1/(2y). Integrate to get y^2 = x + c and use (4,3) to find c = 5, yielding y^2 = x + 5.

Derive the differential equation dy/dx = lambda y/x from the tangent slope condition and solve to obtain the curve y = c x^lambda.