consider the Riccati equation y'=p(x)+q(x)y+r(x)y^2. If a particular solution y1(x), show that the general solution y(x)...
consider the Riccati equation
y'=p(x)+q(x)y+r(x)y^2.
If a particular solution y1(x), show that the general solution y(x) has the from y(x)=y1(x)+z(x);
where z(x) is the solution of the bernoulli equation:
z'-(q+zry1)z=rz^2
Use this technique to find the general solution of the equation,
y'=y/x+x^3y^2-x^5.
(Hint: Verify that y1(x)=x is a particular solution)
Homework Answers
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consider the Riccati equation
y'=p(x)+q(x)y+r(x)y^2.
If a particular solution y1(x), show that the general solution
y(x)...
LEl equation. 8. Consider the equation y" + xy' + y = o. a. Pind its general solution y E cnx in the form y Ayx)By2) where y1 and y2 are power series b. Use the ratio test to verify that...
LEl equation. 8. Consider the equation y" + xy' + y = o. a. Pind its general solution y E cnx in the form y Ayx)By2) where y1 and y2 are power series b. Use the ratio test to verify that the two series y, and yp converge for all x. Write out the theozn in the book both series would converge. 2 , and use this fact c. Show that y,(x) is the series expansion of e , to·ind...
Find the general solution of the dierential equation where y = x^2 is a particular solution...
Find the general solution of the dierential equation where y =
x^2 is a particular solution
2. Find the general solution of the differential equation where y = x2 is a particular solution (1 – xº)y' – 2x + x²y + y2 = 0
A Bernoulli differential equation is one of the form dydx+P(x)y=Q(x)yn. Observe that, if n=0 or 1,...
A Bernoulli differential equation is one of the form dydx+P(x)y=Q(x)yn. Observe that, if n=0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u=y1−n transforms the Bernoulli equation into the linear equation dudx+(1−n)P(x)u=(1−n)Q(x). Use an appropriate substitution to solve the equation y′−5xy=y5x7, and find the solution that satisfies y(1)=1. y(x)=
(Generalized Riccati Equation) Let po, Pi, Pp2 T-R be continous functions defined on an interval I of R. Then the 1st-o...
(Generalized Riccati Equation) Let po, Pi, Pp2 T-R be continous functions defined on an interval I of R. Then the 1st-order differential equations of the type is called generalized Riccati equations. It is another nonlinear ordinary differential equation (a) Suppose, P2 differentiable and P2メ0 on I. By using the Ansatz u(z) :-y(r) P2(x) T, for every z where y is a solution of (2), develope a method to solve the equation (2). Describe in brief steps your method. Hint: The...
(1 point) A Bernoulli differential equation is one of the form dete+ P(x)y= Q(2)y". Observe that,...
(1 point) A Bernoulli differential equation is one of the form dete+ P(x)y= Q(2)y". Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = yl-n transforms the Bernoulli equation into the linear equation am + (1 – n)P(x)u = (1 – n)Q(x). Use an appropriate substitution to solve the equation and find the solution that satisfies y(1) = 1. y(x) =
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the s...
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the series converges for any E (-8,s), s > 0 Substitute the series solution y in to the differential equation and simplify the terms to obtain an expression of the form 1-1 where f(r) is a polynomial of degree 2. (c) Determine the values of r....
Find the general solution of the equation: y'' + 5y = 0 Find the general solution...
Find the general solution of the equation:
y'' + 5y = 0
Find the general solution of the equation and use Euler’s
formula to place the solution in terms of trigonometric
functions:
y'''+y''-2y=0
Find the particular solution of the equation:
y''+6y'+9y=0
where
y1=3
y'1=-2
Part 2: Nonhomogeneous
Equations
Find the general solution of the equation using the method of
undetermined coefficients:
Now find the general solution of the equation using the method
of variation of parameters without using the formula...
Bonus (Abel's formula) a) Show that if y1 and y2 are solutions to the differential equation...
Bonus (Abel's formula) a) Show that if y1 and y2 are solutions to the differential equation y"p(t)y(t)y 0 where p and q are continuous on an interval I, then the Wronskian of y and y2, W(y1,y2) (t) is given by - Sp(t)dt ce W(y1, y2)(t) where c depends on y and y2 (b) Use Abel's formula to find the Wronskian of two solutions to the differential equation ty"(t 1)y 3y 0 Do not solve the differential equation
(Q3) Consider the equation: y′ = y1/3, y(0) = 0 . (a)Does the above IVP have...
(Q3) Consider the equation:
y′ = y1/3, y(0) = 0
. (a)Does the above IVP have any solution?
(b)Is the solution unique?
(c)Interpret your results in light of the theorem of existence
and uniqueness.
(Q3) Consider the equation: y' = y1/3, y(0) = 0 . (a)Does the above IVP have any solution? (b) Is the solution unique? (c)Interpret your results in light of the theorem of existence and uniqueness. (Q4) Solve the following IVP and find the interval of validity:...
(15 points) A Bernoulli differential equation is one of the form dy dar + P(x)y= Q(x)y"....
(15 points) A Bernoulli differential equation is one of the form dy dar + P(x)y= Q(x)y". Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u=yl-n transforms the Bernoulli equation into the linear equation du + (1 - n)P(x) = (1 - nQ(x). dx Use an appropriate substitution to solve the equation xy' +y=2xy? and find the solution that satisfies y(1) = 1.
LEl equation. 8. Consider the equation y" + xy' + y = o. a. Pind its general solution y E cnx in the form y Ayx)By2) where y1 and y2 are power series b. Use the ratio test to verify that the two series y, and yp converge for all x. Write out the theozn in the book both series would converge. 2 , and use this fact c. Show that y,(x) is the series expansion of e , to·ind...
Find the general solution of the dierential equation where y =
x^2 is a particular solution
2. Find the general solution of the differential equation where y = x2 is a particular solution (1 – xº)y' – 2x + x²y + y2 = 0
(Generalized Riccati Equation) Let po, Pi, Pp2 T-R be continous functions defined on an interval I of R. Then the 1st-order differential equations of the type is called generalized Riccati equations. It is another nonlinear ordinary differential equation (a) Suppose, P2 differentiable and P2メ0 on I. By using the Ansatz u(z) :-y(r) P2(x) T, for every z where y is a solution of (2), develope a method to solve the equation (2). Describe in brief steps your method. Hint: The...
(1 point) A Bernoulli differential equation is one of the form dete+ P(x)y= Q(2)y". Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = yl-n transforms the Bernoulli equation into the linear equation am + (1 – n)P(x)u = (1 – n)Q(x). Use an appropriate substitution to solve the equation and find the solution that satisfies y(1) = 1. y(x) =
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the series converges for any E (-8,s), s > 0 Substitute the series solution y in to the differential equation and simplify the terms to obtain an expression of the form 1-1 where f(r) is a polynomial of degree 2. (c) Determine the values of r....
Find the general solution of the equation:
y'' + 5y = 0
Find the general solution of the equation and use Euler’s
formula to place the solution in terms of trigonometric
functions:
y'''+y''-2y=0
Find the particular solution of the equation:
y''+6y'+9y=0
where
y1=3
y'1=-2
Part 2: Nonhomogeneous
Equations
Find the general solution of the equation using the method of
undetermined coefficients:
Now find the general solution of the equation using the method
of variation of parameters without using the formula...
Bonus (Abel's formula) a) Show that if y1 and y2 are solutions to the differential equation y"p(t)y(t)y 0 where p and q are continuous on an interval I, then the Wronskian of y and y2, W(y1,y2) (t) is given by - Sp(t)dt ce W(y1, y2)(t) where c depends on y and y2 (b) Use Abel's formula to find the Wronskian of two solutions to the differential equation ty"(t 1)y 3y 0 Do not solve the differential equation
(Q3) Consider the equation:
y′ = y1/3, y(0) = 0
. (a)Does the above IVP have any solution?
(b)Is the solution unique?
(c)Interpret your results in light of the theorem of existence
and uniqueness.
(Q3) Consider the equation: y' = y1/3, y(0) = 0 . (a)Does the above IVP have any solution? (b) Is the solution unique? (c)Interpret your results in light of the theorem of existence and uniqueness. (Q4) Solve the following IVP and find the interval of validity:...
(15 points) A Bernoulli differential equation is one of the form dy dar + P(x)y= Q(x)y". Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u=yl-n transforms the Bernoulli equation into the linear equation du + (1 - n)P(x) = (1 - nQ(x). dx Use an appropriate substitution to solve the equation xy' +y=2xy? and find the solution that satisfies y(1) = 1.
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