1) Find the general solution of the given differential equation
a) \(y^{\prime \prime}+2 y^{\prime}-3 y=0\),
b) \(y^{\prime \prime}+3 y+2 y=0\),
c) \(4 y^{\prime \prime}-9 y=0\),
d) \(y^{\prime \prime}-9 y^{\prime}+9 y=0\).
2) Find the solution of the given initial value problem and describe the behavior of solution as \(t \rightarrow+\infty\)
$$ y^{\prime \prime}+4 y^{\prime}+3 y=0, \quad y(0)=2, y^{\prime}(0)=-1 $$
3) Find a differential equation whose general solution is \(y=c_{1} e^{2 t}+c_{2} e^{-3 t}\).
1.Find a general solution to the given differential equation. 21y'' + 8y' - 5y = 0 A general solution is y(t) = _______ .2.Solve the given initial value problem. y'' + 3y' = 0; y(0) = 12, y'(0)= - 27 The solution is y(t) = _______ 3.Find three linearly independent solutions of the given third-order differential equation and write a general solution as an arbitrary linear combination of them z"'+z"-21z'-45z = 0 A general solution is z(t) = _______
2. Differential equations and direction fields (a) Find the general solution to the differential equation y' = 20e3+ + + (b) Find the particular solution to the initial value problem y' = 64 – 102, y(0) = 11. (e) List the equilibrium solutions of the differential equation V = (y2 - 1) arctan() (d) List all equilibrium solutions of the differential equation, and classify the stability of each: V = y(y - 6)(n-10) (e) Use equilibrium solutions and stability analysis...
D.E.
(1) y Find the general solution of the differential equation ay - 25 y' + 25 y = 0. (2) Find the particular solution of the initial-value problem y .+ y - 2 y = 0; y(O) = 5, y (0) - - 1 (3) Find the general solution of the differential equation - NO OVERLAP! y. - 3 y - y + 3 y = 54 x - 3e 2x (4) Find the general solution of the differential...
1.- The given family of solutions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial value problem (a) y = cie" + c2e-, 2€ (-0,00) y" - y = 0, y(0) = 0, 10) = 1 y=cles + cze-, 1€ (-00,00) y" – 3y – 4y = 0, y(0) = 1, y(0) = 2 Cl2 + 2x log(x), t (0, x) ry" – ry'...
5.Given that \(y=x\) is a solution of \(\left(x^{2}-x+1\right) y \prime \prime-\left(x^{2}+x\right) y \prime+(x+1) y=0\), a linearly independent solution obtained by reducing the order is given by\(y=e^{x}(x+1)\)\(y=e^{x}(x-1)\)None of them\(y=x^{2} e^{x}\)\(y=x e^{x}\)6. If the functions y = x and y = xex are linearly independent solutions of the non-homogeneous second-order linear differential equation with variable coefficients second-order linear differential equation with variable coefficients\(x^{2} y \prime \prime-x(x+2) y \prime+(x+2) y=x^{3}\), its general solution is given byNone of them\(y=C_{1}+C_{2} x e^{x}+x^{2}\)\(y=C_{1} x^{2}+C_{2} x e^{x}-x^{3}\)\(y=C_{1} x+C_{2}...
The indicated function \(y_{1}(x)\) is a solution of the given differential equation. Use reduction of order or formula (5) in Section 4.2,$$ y_{2}=y_{1}(x) \int \frac{e^{-\int P(x) d x}}{y_{1}^{2}(x)} d x $$as instructed, to find a second solution \(y_{2}(x)\).$$ y^{\prime \prime}-y=0 ; \quad y_{1}=\cosh x $$
1. Consider the differential equation" = y2 - 4y - 5. a) Find any equilibrium solution(s). b) Create an appropriate table of values and then sketch (using the grid provided) a direction field for this differential equation on OSIS 3. Be sure to label values on your axes. c) Using the direction field, describe in detail the behavior of y ast approaches infinity. 2. Short answer: State whether or not the differential equation is linear. If it is linear, state...
(a) Find the general solution of the following second order linear differential equation given that y1 = t is known to be a solution: t2y" - (t2 + 2t) y' + (t + 2)y = 0, t> 0. (b) Find the particular solution given that y(1) = 7 and y'(1) = 4.
Find the general solution of the following differential
equation: (1) ?′′ + 5?′ + 6? = 2????*?^? (2) ?′′ + 2?′ + ? = ? +
?e^(-t).
(please solve Question No.7 only)
7. (30 points) Find the general solution of the following differential equation: (1) y" + 5y' + 6y = 2etsint (2) y" + 2y + y=t+te-t 8. (10 points) Use the method of variation of parameters to find a particular solution of y" + y = 1/sin (t),...
1) Write a differential equation describing this system. This means find the equation of the line in the graph. df ar= 1x-80 2) Find the general solution to this differential equation. Find the function f(x) whose derivative is the equation of the line graphed. The solution is: f(r) -.5x 2-80x 3) Now given that function f(x) includes the point (0, 100) find the exact solution of the differential equation found in 1). In addition to general solution you will have...