[If the second Picard approximation of the following initial value problem is y2(x), find y2(1)]. Sy' = 10 + y, ly(0) = 16. %3D 51 O 54 O 55 52 O 53

Q: 7 1. 2ydx+x²dy = - dx; y = - when x =1 In 2 2

A: Given:- 2y dx +x2 dy =-dx  To find:- Solution of Differential equation at y = 7/2 when x =1/ln(2)

Q: y" + 4.5y' = 0 %3D

A:

Q: 3. Find the dy/dx of * y? + 2x – 3y = 0

A: The given function,         y2+2x-3y=0

Q: O XsinCyldx +(x^t)cos(y) dy=0

A: We have  xsinydx+x2+1cosydy = 0  x2+1cosydy=-xsinydx cosydysiny=-xdxx2+1

Q: 22. (e¹ + y) dx + (2 + x + ye³) dy = 0, y(0) = 1

A: Given ex+ydx+2+x+yeydy=0 and y0=1

Q: 2y dx (1-y)' 7

A:

Q: 4. Show that 2 s w (b) L(t sinh (wt)) (s² – w²)2 -

A:

Q: d*y 4. Find dx 1 if y=x' + 5 +13 2x? 3x 5

A:

Q: x sin? dy = (ysin? – x) dx %3D

A:

Q: d. y= x² + y x + cos y

A:

Q: 2xydy + (y- 3x-)dx = 0;y(3) = 2 O xy² - x³ = c O x³ 2 – xy² = -15 Ox* - xy? = 15 O x³- xy? = c

A:

Q: 3. (D² — 2D + 1) у %3 е* In a

A:

Q: Find the area of each circle, given the radius r or the diameter d. Leave your answer in tems of t.

A: See the detailed solution below.

Q: 4. Find d²y/dx² for the following functions: c) y = x²(x³ – 1)5 d) y = (1 – x'/2)¯ 1

A:

Q: 6. 2xy dx + (y? – x²) dy = 0

A: The solution is given as

Q: What does is mean for two functions f(x) and g(x) to be orthogonal

A:

Q: d²y 7. – 4y=-6e', y(0)=3; y'(0)=4 di? 2/2

A:

Q: 1. e-st cosh t dt 2x +4 2. 2x2 + 21x + 34

A:

Q: dy If x³+y³ = 1, what is -? dx² D OC O 22(23 ys 2x(2 − y) y³ 2x y5

A:

Q: (4) (b) 10tsin(5t²- n/4)dt (4) (c) S 12(x³ + 2)ex¹+8x dx

A: We have to evaluate the integrals.

Q: 1 1. (2e²* +4y)dx+(4x + ₂)dy = 0

A: For a differential equation  ∫Mdx + ∫Ndy = 0 for differential equation to be exact ∂M∂y = ∂N∂x

Q: Show that y = x° + 3x +1 Satisfies y" + xy" – 2y' = 0.

A:

Q: o(e) = La t -Va V1 + t²

A:

Q: y 4 Find Pr(3 < Y < 5), correct to four decimal places.

A: There are two types of variables: Continuous variables and discrete variables. Continuous variables…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.