Question

First-Order ODE

(a) .Find the general solution of the following ODE:

  

2Ꮖy 2 Ꭸ , . + <+5 12 ,

(b). Find the general solution (for x > 0) of the ODE :

  

Hint: try the change of variables u ≜ x, v ≜ y/x.

(c). Find the solution to the ODE

that satisfies y(2) = 15.

  Hint: Try separation of variables. For integration, try partial fraction decomposition.

Answer 1

(1):-

( dy(x))/( dx) - (2 x y(x))/(x^2 + 5) = x^2 + 5
Let μ(x) = e^( integral-(2 x)/(x^2 + 5) dx) = 1/(x^2 + 5).
Multiply both sides by μ(x):
(( dy(x))/( dx))/(x^2 + 5) - ((2 x) y(x))/(x^2 + 5)^2 = -(-x^2 - 5)/(x^2 + 5)
Substitute -(2 x)/(x^2 + 5)^2 = d/( dx)(1/(x^2 + 5)):
(( dy(x))/( dx))/(x^2 + 5) + d/( dx)(1/(x^2 + 5)) y(x) = -(-x^2 - 5)/(x^2 + 5)
Apply the reverse product rule f ( dg)/( dx) + g ( df)/( dx) = d/( dx)(f g) to the left-hand side:
d/( dx)(y(x)/(x^2 + 5)) = -(-x^2 - 5)/(x^2 + 5)
Integrate both sides with respect to x:
integral d/( dx)(y(x)/(x^2 + 5)) dx = integral-(-x^2 - 5)/(x^2 + 5) dx
Evaluate the integrals:
y(x)/(x^2 + 5) = x + c_1, where c_1 is an arbitrary constant.
Divide both sides by μ(x) = 1/(x^2 + 5):
Answer: |
| y(x) = (x + c_1) (x^2 + 5)

(2):-

x ( dy(x))/( dx) - y(x)^2 - y(x) = x^2:
Let y(x) = x v(x), which gives ( dy(x))/( dx) = x ( dv(x))/( dx) + v(x):
x (x ( dv(x))/( dx) + v(x)) - x^2 v(x)^2 - x v(x) = x^2
Simplify:
x^2 (( dv(x))/( dx) - v(x)^2) = x^2
Solve for ( dv(x))/( dx):
( dv(x))/( dx) = v(x)^2 + 1
Divide both sides by v(x)^2 + 1:
(( dv(x))/( dx))/(v(x)^2 + 1) = 1
Integrate both sides with respect to x:
integral(( dv(x))/( dx))/(v(x)^2 + 1) dx = integral1 dx
Evaluate the integrals:
tan^(-1)(v(x)) = x + c_1, where c_1 is an arbitrary constant.
Solve for v(x):
v(x) = tan(x + c_1)
Substitute back for y(x) = x v(x):
Answer: |
| y(x) = x tan(x + c_1)

(3):-

Divide both sides by y(x)^2 + y(x) - 20:
(( dy(x))/( dx))/(y(x)^2 + y(x) - 20) = 1
Integrate both sides with respect to x:
integral(( dy(x))/( dx))/(y(x)^2 + y(x) - 20) dx = integral1 dx
Evaluate the integrals:
1/9 log(-y(x) + 4) - 1/9 log(y(x) + 5) = x + c_1, where c_1 is an arbitrary constant.
Solve for y(x):
y(x) = (-5 e^(9 (x + c_1)) + 4)/(e^(9 (x + c_1)) + 1)
Solve for c_1 using the initial conditions:
Substitute y(2) = 15 into y(x) = (-5 e^(9 (x + c_1)) + 4)/(e^(9 (x + c_1)) + 1):
(-5 e^(9 (c_1 + 2)) + 4)/(e^(9 (c_1 + 2)) + 1) = 15
Solve the equation:
c_1 = 1/9 (-18 + i π - log(20/11))
Substitute c_1 = 1/9 (-18 + i π - log(20/11)) into y(x) = (-5 e^(9 (x + c_1)) + 4)/(e^(9 (x + c_1)) + 1):
Answer: |
| y(x) = (55 e^(9 x) + 80 e^18)/(-11 e^(9 x) + 20 e^18)