Differential Equations A Problem Based Approach

Explore how ordinary differential equations model change through derivatives, with examples from engineering and biology, and introduce linear first-order differential equations and their general solution c e^{-t}.

Learn to solve linear first order differential equations, distinguish homogeneous from inhomogeneous cases, and apply the integrating factor method to rewrite the equation as the derivative of a product.

Use the integrating factor sec x to transform cos x y' + sin x y equals zero. Show that y equals C cos x is the solution.

Use the integrating factor e^x to solve y' + y = x e^x, convert to derivative of y e^x, and apply integration by parts.

Apply integration by parts with f = x and g' = e^{2x}, use the chain rule to get g = e^{2x}/2, and simplify toward solving for y.

Solve the differential equation y' + y = x e^x using the integrating factor e^x, and obtain y(x) = e^x(x/2 + 1/4) + c e^{-x}.

Solve a first-order differential equation by separable methods, turning the right-hand side into a product of functions of x and y, then separate, integrate, and obtain the general solution.

Solve a separable first-order ode by separating variables, integrating to relate ln(x^2+1) with 1/y, and obtain the solution y = 1 / (1/2 ln(x^2+1) + c).

Solve the initial value problem dy/dx equals this expression with y(0)=2, determine the constant c, and obtain a unique solution.

Apply separation of variables to solve the initial value problem for the logistic equation. The solution is y(x)=d e^x/(1+d e^x), with y(0)=2 determining d.

Review the differential equation under a new initial condition, determine the constant d, and show the solution tends to one as x grows, illustrating asymptotic behavior.

Explore solving homogeneous first order odes by applying the v = y/x substitution and y = v(x)·x, and verify that the right-hand side is homogeneous of degree zero.

Demonstrates, in differential equations, that the right-hand side f(x,y) is homogeneous of degree zero by substituting lambda x and lambda y and simplifying to recover f(x,y).

Apply homogeneous first-order ODE theory by setting v = y/x. Let y = v x, so y' = v' x + v, yielding a separable equation in x and v.

Solve a homogeneous differential equation via v = y/x, obtain a separable equation, integrate to sin(v) = ln|x| + C, then recover y = x arcsin(ln|x| + C).

Solve an ode by substituting v = y/x and separating variables. Compute sin v = ln|x| + C, then y = x arcsin( ln|x| + C ).

Develop Bernoulli ODEs by using an integrating factor, setting v = y mu, deriving a separable equation v' = g(x) v^{α} mu(x)^{1-α}, and recovering y from v.

Rewrite the equation as a Bernoulli ODE, apply the integrating factor, and use v = y x^{-2} to obtain a separable equation for v.

Apply an integrating factor to the Bernoulli ODE, set v = y times the integrating factor to obtain a separable equation, and integrate to express y in terms of x.

Solve a Bernoulli ODE by substituting v = y x^-2, yielding y^3 = -3/4 x^2 + C x^6, and find y(x) = cube root(-3/4 x^2 + C x^6).

Solve an initial value problem by rewriting the ode as a Bernoulli equation and using an integrating factor. Determine the constant from the initial condition and finish the solution.

Solve a Bernoulli ode by applying an integrating factor e^{-x}, transform with v = y e^{-x}, and use separation of variables to complete the exercise.

solve a Bernoulli ODE by substituting for y, multiply by e^{-x}, and reduce to separation of variables, using integration by parts.

Apply integration by parts to evaluate the integral x e^{4x}, obtain e^{4x}/16 plus a constant, and relate y to v via y = v e^x, preparing to solve for y.

Complete the computation of this Bernoulli ODE integral by transforming to y^−4 and obtaining y^4 as the reciprocal of the right-hand expression, yielding an implicit solution.

Explore linear homogeneous equations with constant coefficients and derive the characteristic equation using the trial solution y = e^{lambda x}, preparing for algebraic multiplicity.

Explore how algebraic multiplicity of roots in the characteristic polynomial yields additional solutions for constant-coefficient linear ODEs, by multiplying e^{λx} by powers of x to complete the fundamental solution set.

this lecture explains that complex non-real roots occur in conjugate pairs when coefficients are real, and hints at deriving real solutions via sine and cosine from real and imaginary parts.

From a complex root alpha plus i beta with multiplicity s, derive real solutions e^{alpha x} cos(beta x) and e^{alpha x} sin(beta x) with x^k up to k=s-1.

Solve third-order homogeneous linear differential equation with constant coefficients by factoring its characteristic polynomial, yielding roots 2, 7, -4, and y = c1 e^{2x} + c2 e^{7x} + c3 e^{-4x}.

Analyze a linear inhomogeneous second-order ODE, showing y1 minus y2 solves the homogeneous equation and a y1 plus b y2 solves the original equation when a plus b equals one.

Solve a separable first-order ode by separating variables and integrating. Obtain y = 1 / ( (1/2) ln(x^2+1) + C ), using the 1/y^2 integral.

Solve a fourth-order constant-coefficient ode with complex roots, derive the characteristic equation, and obtain the fundamental set {cos x, x cos x, sin x, x sin x} for general solution.

Solve second order linear homogeneous odes with constant coefficients using the characteristic equation, determine real, repeated, or complex roots, and derive the general solution from y = e^{lambda x}.

Learn to form the characteristic equation from a linear differential equation with e^{r x}, divide by e^{r x}, and classify roots via b^2 - 4ac into real, repeated, or complex.

Explore delta in ay''+by'+cy=0: real distinct roots yield y=c1 e^{r1 x}+c2 e^{r2 x}; repeated root yields y=c1 e^{r x}+c2 x e^{r x}; complex roots yield y=c1 cos(alpha x)+c2 sin(beta x).

Solve the worksheet on second order linear homogeneous ODEs ay'' + by' + cy = 0, finding the characteristic equation, roots, the general solution, and initial conditions when given.

solve the differential equation y'' + y' = 0 via the characteristic equation r(r+1)=0, yielding y(x) = c1 + c2 e^x, as part of problem-based learning.

Solve y'' + y' - 6y = 0 using the characteristic equation, find roots 2 and -3, and express the general solution as a linear combination of e^{2x} and e^{-3x}.

Solve the differential equation with a double real root using the characteristic equation, form the fundamental set {e^{-3x}, x e^{-3x}}, apply y(0)=1 and y'(0)=-5 to get y(x)=e^{-3x}-2x e^{-3x}.

From quadratic with a=b=c=1, obtain complex roots lambda = -1/2 ± i sqrt(3)/2. Form fundamental solutions e^{-x/2} cos(beta x) and e^{-x/2} sin(beta x) and general solution as a linear combination.

Apply the variation of parameters to second-order linear inhomogeneous ODEs, using homogeneous solutions y1, y2, the wronskian, and formulas for u1, u2 to obtain the particular and general solutions.

Use variation of parameters on y''+4y=0, derive y1 and y2, compute u1' and u2' with w = y1 y2' - y1' y2, and form the general solution.

Apply variation of parameters to a second-order equation, obtain y_h = c1 cos 2x + c2 sin 2x, and compute the Wronskian to derive u1' and u2' for g(x)=tan 2x.

Finish solving the differential equation by applying variation of parameters to find u1 and u2 for the particular solution y_p, then combine with the homogeneous solution y_h.

Explore systems of first-order differential equations, where x(t) is a vector with components x1..xn, with A(t) acting on x and an optional g, and homogeneous cases when g = 0.

Rewrite the system in matrix notation with a constant coefficient matrix A, viewing x' as the derivative of x for a homogeneous problem, preparing for eigenvalues, eigenvectors, and diagonalization.

Convert a linear nth-order ODE to a first-order system by defining y0 = y, y1 = y', ..., y(n-1) = y^(n-1), and express y' = A y.

Discover that the solution set of a homogeneous first-order linear ode system x' = A x forms an n-dimensional vector space.

Explore solving systems of first-order odes with constant coefficients by seeking x = v e^{λ t} and deriving the eigenvalue–eigenvector condition A v = λ v.

Explore eigenvalues and eigenvectors of a symmetric matrix A, show it is diagonalizable, identify eigenvalues 2 and -1 with eigenvector 111, and outline solving for eigenvectors via Gaussian elimination.

Find eigenvectors for eigenvalue 2 by Gaussian elimination of A-2I, deducing x=z, y=z, z real, yielding eigenspace span{[1,1,1]} and solution x1(t)=e^(2t)[1,1,1].

Find eigenvectors for lambda = -1 by forming A minus lambda I and applying Gaussian elimination. Express the eigenspace as the span of (-1,1,0) and (-1,0,1) and obtain e^{-t}-weighted solutions.

Solve a first-order ode system with complex eigenvalues 1 ± i, determine the eigenvector, and derive the general solution as e^t times a linear combination of sine and cosine terms.

Apply variation of parameters to inhomogeneous ODEs. Solve homogeneous problem and express general solution as matrix combination of x1 to xn, with the particular solution XP defined by vector c.

Apply variation of parameters to inhomogeneous systems using the fundamental matrix X of homogeneous solutions to derive c'(t) = X^{-1}(t) g(t) and form a particular solution XP = X c.

Multiply by the inverse of x, guaranteed invertible by the linear independence of x1 to xn, to turn C' into C. Integrate both sides to obtain the particular solution C.