**Question 1:** Show that y=be^{x}+ce^{2x} is the solution of the differential equation.

**d ^{2} y/dx^{2}-3(dy/dx)+2y=0**

**Solution:**

y=be

^{x}+ce^{2x}(i)Differentiating equation (i)w.r.t x,

dy/dx=be

^{x}+2ce^{2x}dy/dx=be

^{x}+2ce^{2x}(ii)Again, differentiating equation (ii)w.r.t x,

d

^{2}y/dx^{2 }=be^{x}+4ce^{2x}(iii)we have,

d

^{2}y/dx^{2}-3(dy/dx)+2y=0 (iv)Putting the values ofd

^{2 }y/dx^{2}anddy/dx in equation (iv)=be

^{x}+4ce^{2x}-3(be^{2x}+2ce^{2x})+2(be^{x}+ce^{2x})=3be

^{x}-3be^{x}+6ce^{2x}-6ce^{2x}=0

So,d

^{2}y/dx^{2}-3(dy/dx)+2y=0

**Question 2: **Verify that y=4sin3x** **is a solution of the differential equation.

**d ^{2}y/dx^{2}+9y=0 **

**Solution:**

y=4sin3x (i)

Differentiating equation (i)w.r.t x,

dy/dx=(4)(3)cos3x (ii)

Again differentiating equation (ii)w.r.t x,

d

^{2}y/dx^{2 }=-(12)(3)sin3xd

^{2}y/dx^{2}=-(9)(4sin3x)d

^{2}y/dx^{2}=-9y (Since y=4sin3x)d

^{2}y/dx2+9y=0So, d

^{2}y/dx^{2}+9y=0

**Question 3: **Show that y=ae^{2x}+be^{−x}** **is a solution of the differential equation.

**d ^{2}y/dx^{2}-dy/dx-2y=0**

**Solution:**

y=ae

^{2x}+be^{−x}(i)Differentiating equation (i)w.r.t x,

dy/dx=2ae

^{2x}-be^{-x}(ii)Again differentiating equation (ii)w.r.t x,

d

^{2}y/dx^{2}=4ae^{2x}+be^{-x}(iii)we have,

d

^{2}y/dx^{2}-dy/dx-2y (iv)Putting the values ofandin equation (iv)

=4ae

^{2x}+be^{-x}-(2ae^{2x}-be^{-x})-2(ae^{2x}+be^{−x})=4ae

^{2x}-4ae^{2x}+be^{−x}-be^{−x})=0

**Question 4: **Show that the function, y=Acosx-Bsinx is a solution of the differential equation.

**d ^{2}y/dx^{2}+y=0**

**Solution:**

y=Acosx-Bsinx (i)

Differentiating equation (i)w.r.t x,

dy/dx=-Asinx-Bcosx (ii)

Again differentiating equation (ii)w.r.t x,

d

^{2}y/dx^{2}=-Acosx+Bsinxd

^{2}y/dx^{2}=-(Acosx-Bsinx)d

^{2}y/dx^{2}+(Acosx-Bsinx)=0d

^{2}y/dx^{2}+y=0 (since y=Acosx-Bsinx)

**Question 5:** Show that the function, y=Acos2x-Bsin2x is a solution of the differential equation.

**d ^{2}y/dx^{2} + 4y = 0**

**Solution:**

y=Acos2x-Bsin2x (i)

Differentiating equation (i)w.r.t x,

dy/dx=-2Asin2x-2Bcos2x (ii)

Again differentiating equation (ii)w.r.t x,

d

^{2}y/dx^{2}=-4Acos2x+4Bsin2xd

^{2}y/dx+4(Acos2x-Bsin2x)=0^{2}d

^{2}y/dx^{2}+4y=0 (since y=Acos2x-Bsin2x)

**Question 6: **Show that, y=Ae^{Bx }is the solution of the differential equation.

**d ^{2}y/dx^{2}=(1/y)(dy/dx)^{2}**

**Solution:**

y=Ae

^{Bx}(i)Differentiating equation (i)w.r.t x,

dy/dx=ABe

^{Bx}(ii)Again differentiating equation (ii)w.r.t x,

d

^{2}y/dx^{2}=AB^{2}e^{bx}d

^{2}y/dx^{2}=(ABe^{bx})^{2}/(Ae^{Bx})d2y/dx2=(1/y)(dy/dx)

^{2}

**Question 7:** Verify that y= (x/a)+b is the solution of the differential equation.

**d ^{2}y/dx^{2}+(2/x)(dy/dx)^{2}=0**

**Solution:**

y= (x/a)+b (i)

Differentiating equation (i)w.r.t x,

dy/dx=-(a/x

^{2}) (ii)Again differentiating equation (ii)w.r.t x,

d

^{2}y/dx^{2}=+(2a/x^{3})d

^{2}y/dx^{2}=-(-2/x)(a/x^{2})d

^{2}y/dx^{2}+(2/x)(dy/dx)=0

**Question 8:** Verify that y^{2}=4ax is the solution of the differential equation.

**x(dy/dx)+y(dx/dy)=y**

**Solution:**

y

^{2}=4ax (i)Differentiating equation (i)w.r.t x,

2y(dy/dx)=4a

dy/dx=(2a/y)

we have,

x(dy/dx)+y(dx/dy)

=x(2a/y)+y(y/2a)

=(4xa+y

^{2})/2y=(2y

^{2}/2y)=y

**Question 9:** Show that Ax^{2}+By^{2}=1 is the solution of the differential equation.

**Solution:**

Ax

^{2}+By^{2}=1 (i)Differentiating equation (i)w.r.t x,

2Ax+2By(dy/dx)=0

2Ax=-2By(dy/dx)

y(dy/dx)=-(Ax/B) (ii)

Again differentiating equation (ii)w.r.t x,

(dy/dx)

^{2}+y(d^{2}y/dx^{2})=-(A/B)(dy/dx)

^{2}+y(d^{2}y/dx^{2})=-(y/x)(dy/dx)

**Question 10:** Show that y=ax^{3}+bx^{2}+cis the solution of the differential equation.

**(d ^{3}y/dx^{3})=6a**

**Solution:**

We have,

y=ax

^{3}+bx^{2}+c (i)Differentiating equation (i)w.r.t x,

(dy/dx)=3ax

^{2}+2bx (ii)Again differentiating equation (ii)w.r.t x,

(d

^{2}y/dx^{2})=6ax (iii)Again differentiating equation (iii)w.r.t x,

(d

^{3}y/dx^{3})=6a

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