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- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/08%3A_Linear_Systems_of_Differential_Equations/8.06%3A_Constant_Coefficient_Homogeneous_Systems_IIIWe now consider the system y′=Ay , where A has a complex eigenvalue λ=α+iβ with β≠0 . We continue to assume that A has real entries, so the characteristic polynomial of A has real coefficient...We now consider the system y′=Ay , where A has a complex eigenvalue λ=α+iβ with β≠0 . We continue to assume that A has real entries, so the characteristic polynomial of A has real coefficients. This implies that λ=α−iβ is also an eigenvalue of A .
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/05%3A_Applications_of_Linear_Second_Order_Equations/5.03%3A_The_RLC_CircuitIn this section we consider the RLC circuit, which is an electrical analog of a spring-mass system with damping.
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/zz%3A_Back_Matter/10%3A_Index
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/04%3A_Linear_Higher_Order_Differential__Equations/4.01%3A_Second_Order_Homogeneous_Linear_EquationsThis section is devoted to the theory of homogeneous linear equations.
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/06%3A_Laplace_Transforms/6.02%3A_The_Inverse_Laplace_TransformThis section deals with the problem of finding a function that has a given Laplace transform.
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/09%3A_Appendices/9.02%3A_Solving_Systems_with_Cramer's_RuleIn this section, we will study two more strategies for solving systems of equations. A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as...In this section, we will study two more strategies for solving systems of equations. A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a square matrix to determine whether there is a solution to the system of equations. Cramer’s Rule to solve a system of equations in two & three variables.
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/03%3A_Applications_of_First_Order_Equations/3.05%3A_Applications_to_CurvesThis section deals with applications of differential equations to curves.
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/04%3A_Linear_Higher_Order_Differential__Equations/4.07%3A_Undetermined_Coefficients_for_Second_Order_Equations_(Trigonometry_Forcing)In this section, we use the Method of Undetermined Coefficients to find solutions to the constant coefficient equation ay''+by'+cy=exp{λx}(P(x) cos ω x + Q(x) sin ω x) where λ and ω are real numbers, ...In this section, we use the Method of Undetermined Coefficients to find solutions to the constant coefficient equation ay''+by'+cy=exp{λx}(P(x) cos ω x + Q(x) sin ω x) where λ and ω are real numbers, ω is not zero, and P and Q are polynomials.
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/09%3A_Appendices/9.01%3A_A_Short_ReviewA basic understanding of pre-calculus, calculus, and complex numbers is required for this course. This zero chapter presents a concise review.
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/02%3A_First_Order_Equations/2.05%3A_Transformation_of_Nonlinear_Equations_into_Separable_Equations/2.5E%3A_Transformation_of_Nonlinear_Equations_into_Separable_Equations_(Exercises)Based on your observations, find conditions on the positive numbers x0 and y0 such that the initial value problem xyy′=x2−xy+y2,y(x0)=y0, has a unique solution (i) on ...Based on your observations, find conditions on the positive numbers x0 and y0 such that the initial value problem xyy′=x2−xy+y2,y(x0)=y0, has a unique solution (i) on (0,∞) or (ii) only on an interval (a,∞), where a>0? A generalized Riccati equation is of the form y′=P(x)+Q(x)y+R(x)y2. (If R≡−1, (A) is a Riccati equation.) Let y1 be a known solution and y an arbitrary solution of (A).
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/04%3A_Linear_Higher_Order_Differential__Equations/4.09%3A_Reduction_of_Order/4.9E%3A_Reduction_of_Order_(Exercises)10. x2y″ 11. x^2y''-x(2x-1)y'+(x^2-x-1)y=x^2e^x; \quad y_1=xe^x 20. x^2(\ln |x|)^2y''-(2x \ln |x|)y'+(2+\ln |x|)y=0; \quad y_1=\ln |x| ...10. x^2y''+2x(x-1)y'+(x^2-2x+2)y=x^3e^{2x}; \quad y_1=xe^{-x} 11. x^2y''-x(2x-1)y'+(x^2-x-1)y=x^2e^x; \quad y_1=xe^x 20. x^2(\ln |x|)^2y''-(2x \ln |x|)y'+(2+\ln |x|)y=0; \quad y_1=\ln |x| 31. x^2y''-3xy'+4y=4x^4,\quad y(-1)=7,\quad y'(-1)=-8; \quad y_1=x^2 33. (x+1)^2y''-2(x+1)y'-(x^2+2x-1)y=(x+1)^3e^x, \quad y(0)=1,\quad y'(0)=~-1; \quad y_1=(x+1)e^x 34. x^2y''+2xy'-2y=x^2, \quad y(1)={5\over4},\; y'(1)={3\over2}; \quad y_1=x
