Homogeneous equation (y = vx substitution)

6 Use the substitution \(y = v x\) to find the solution of the differential equation $$x \frac { d y } { d x } = y + \sqrt { 9 x ^ { 2 } + y ^ { 2 } }$$ for which \(y = 0\) when \(x = 1\). Give your answer in the form \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), where \(\mathrm { f } ( x )\) is a polynomial in \(x\). [10] \includegraphics[max width=\textwidth, alt={}, center]{114ece0d-558d-4c02-8a77-034b3681cff9-10_51_1648_527_246}

6 Use the substitution \(y = v x\) to find the solution of the differential equation $$x \frac { d y } { d x } = y + \sqrt { 9 x ^ { 2 } + y ^ { 2 } }$$ for which \(y = 0\) when \(x = 1\). Give your answer in the form \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), where \(\mathrm { f } ( x )\) is a polynomial in \(x\). [10] \includegraphics[max width=\textwidth, alt={}, center]{69c540e1-1dad-45a1-9809-7629d16260e0-10_51_1648_527_246}

3. (a) Show that the substitution \(y = v x\) transforms the differential equation $$\frac { d y } { d x } = \frac { 3 x - 4 y } { 4 x + 3 y }$$ into the differential equation $$x \frac { \mathrm {~d} v } { \mathrm {~d} x } = - \frac { 3 v ^ { 2 } + 8 v - 3 } { 3 v + 4 }$$ (b) By solving differential equation (II), find a general solution of differential equation (I). (5)
(c) Given that \(y = 7\) at \(x = 1\), show that the particular solution of differential equation (I) can be written as $$( 3 y - x ) ( y + 3 x ) = 200$$ (5)(Total 14 marks)

12. (a) Use the substitution \(y = v x\) to transform the equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 4 x + y ) ( x + y ) } { x ^ { 2 } } , x > 0$$ into the equation $$x \frac { \mathrm {~d} v } { \mathrm {~d} x } = ( 2 + v ) ^ { 2 }$$ (b) Solve the differential equation II to find \(\boldsymbol { v }\) as a function of \(\boldsymbol { x }\) (c) Hence show that \(\quad y = - 2 x - \frac { x } { \ln x + c }\), where \(c\) is an arbitrary constant, is a general solution of the differential equation I.

1. (a) Show that the substitution \(y = v x\) transforms the differential equation

$$3 x y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x ^ { 3 } + y ^ { 3 }$$ into the differential equation $$3 v ^ { 2 } x \frac { \mathrm {~d} v } { \mathrm {~d} x } = 1 - 2 v ^ { 3 }$$ (b) By solving differential equation (II), find a general solution of differential equation (I) in the form \(y = \mathrm { f } ( x )\). Given that \(y = 2\) at \(x = 1\),
(c) find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = 1\)

4 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } - y ^ { 2 } } { x y }$$

1. Use the substitution \(y = x z\), where \(z\) is a function of \(x\), to obtain the differential equation $$x \frac { \mathrm {~d} z } { \mathrm {~d} x } = \frac { 1 - 2 z ^ { 2 } } { z }$$
2. Hence show by integration that the general solution of the differential equation (A) may be expressed in the form \(x ^ { 2 } \left( x ^ { 2 } - 2 y ^ { 2 } \right) = k\), where \(k\) is a constant.

1 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x ^ { 2 } + y ^ { 2 } } { x y } .$$

1. Use the substitution \(y = u x\), where \(u\) is a function of \(x\), to obtain the differential equation $$x \frac { \mathrm {~d} u } { \mathrm {~d} x } = \frac { 2 } { u } .$$
2. Hence find the general solution of the differential equation (A), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).

1. Show that the substitution \(y = vx\) transforms the differential equation $$3xy^2 \frac{dv}{dx} = v^4 + y^3$$ [I] into the differential equation $$3x v^2 \frac{dv}{dx} = 1 - 2v^3$$ [II] [3]
2. By solving differential equation (II), find a general solution of differential equation (I) in the form \(y = f(x)\). [6]

Given that \(y = 2\) at \(x = 1\),

1. find the value of \(\frac{dy}{dx}\) at \(x = 1\). [2]

1. Show that the substitution \(y = vx\) transforms the differential equation $$\frac{dy}{dx} = \frac{x}{y} + \frac{3y}{x}, x > 0, y > 0$$ (I) into the differential equation \(x\frac{dv}{dx} = 2v + \frac{1}{v}\). (II) (3)
2. By solving differential equation (II), find a general solution of differential equation (I) in the form \(y = f(x)\). (7)

Given that \(y = 3\) at \(x = 1\), (c)find the particular solution of differential equation (I).(2)