The document NCERT Solutions (Part - 2) - Differential Equations JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 12.

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**Question 41: **

**ANSWER : -** The given differential equation is:

Integrating both sides of this equation, we get:

This is the required general solution of the given differential equation.

**Question 42: **

**ANSWER : -** The given differential equation is:

Integrating both sides, we get:

Substituting this value in equation (1), we get:

This is the required general solution of the given differential equation.

**Question 43: **

**ANSWER : -** The given differential equation is:

Integrating both sides, we get:

This is the required general solution of the given differential equation.

**Question 44: **

**ANSWER : -** The given differential equation is:

Integrating both sides, we get:

Substituting this value in equation (1), we get:

This is the required general solution of the given differential equation.

**Question 45: **

**ANSWER : -** The given differential equation is:

Integrating both sides, we get:

Substituting the values of in equation (1), we get:

This is the required general solution of the given differential equation.

**Question 46: **

**ANSWER : -** The given differential equation is:

Integrating both sides, we get:

Comparing the coefficients of *x*^{2} and *x*, we get:

*A* + *B* = 2

*B* + *C* = 1

*A* + *C *= 0

Solving these equations, we get:

Substituting the values of A, B, and C in equation (2), we get:

Therefore, equation (1) becomes:

Substituting C = 1 in equation (3), we get:

**Question 47: **

**ANSWER : -**

Integrating both sides, we get:

Comparing the coefficients of *x*^{2}, *x,* and constant, we get:

Solving these equations, we get

Substituting the values of *A*, *B,* and *C* in equation (2), we get:

Therefore, equation (1) becomes:

Substituting the value of *k*^{2 }in equation (3), we get:

**Question 48: **

**ANSWER : -**

Integrating both sides, we get:

Substituting C = 1 in equation (1), we get:

**Question 49: **

**ANSWER : -**

Integrating both sides, we get:

Substituting C = 1 in equation (1), we get:

*y* = sec *x*

**Question 50: **Find the equation of a curve passing through the point (0, 0) and whose differential equation is.

**ANSWER : -** The differential equation of the curve is:

Integrating both sides, we get:

Substituting this value in equation (1), we get:

Now, the curve passes through point (0, 0).

Substituting in equation (2), we get:

Hence, the required equation of the curve is

**Question 51: For the differential equation find the solution curve passing through the point (1, â€“1).**

**ANSWER : -** The differential equation of the given curve is:

Integrating both sides, we get:

Now, the curve passes through point (1, â€“1).

Substituting C = â€“2 in equation (1), we get:

This is the required solution of the given curve.

**Question 52: Find the equation of a curve passing through the point (0, â€“2) given that at any point on the curve, the product of the slope of its tangent and y-coordinate of the point is equal to the x-coordinate of the point.**

**ANSWER : -** Let *x *and *y* be the *x*-coordinate and *y*-coordinate of the curve respectively.

We know that the slope of a tangent to the curve in the coordinate axis is given by the relation,

According to the given information, we get:

Integrating both sides, we get:

Now, the curve passes through point (0, â€“2).

âˆ´ (â€“2)^{2} â€“ 0^{2} = 2C

â‡’ 2C = 4

Substituting 2C = 4 in equation (1), we get:

*y*^{2} â€“ *x*^{2} = 4

This is the required equation of the curve.

**Question 53: At any point ( x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (â€“4, â€“3). Find the equation of the curve given that it passes through (â€“2, 1).**

**ANSWER : -** It is given that (*x*, *y*) is the point of contact of the curve and its tangent.

The slope (*m*_{1}) of the line segment joining (*x*, *y*) and (â€“4, â€“3) is

We know that the slope of the tangent to the curve is given by the relation,

According to the given information:

Integrating both sides, we get:

This is the general equation of the curve.

It is given that it passes through point (â€“2, 1).

Substituting C = 1 in equation (1), we get:

*y* + 3 = (*x* + 4)^{2}

This is the required equation of the curve.

**Question 54: The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.**

**ANSWER : -** Let the rate of change of the volume of the balloon be *k* (where *k* is a constant).

Integrating both sides, we get:

â‡’ 4Ï€ Ã— 3^{3 }= 3 (*k* Ã— 0 C)

â‡’ 108Ï€ = 3C

â‡’ C = 36Ï€

At *t *= 3, *r* = 6:

â‡’ 4Ï€ Ã— 6^{3} = 3 (*k* Ã— 3 + C)

â‡’ 864Ï€ = 3 (3*k* + 36Ï€)

â‡’ 3*k* = â€“288Ï€ â€“ 36Ï€ = 252Ï€

â‡’ *k* = 84Ï€

Substituting the values of *k* and C in equation (1), we get:

Thus, the radius of the balloon after *t* seconds is.

**Question 55: In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs 100 doubles itself in 10 years (logÂ_{e} 2 = 0.6931).**

**ANSWER : -** Let *p*, *t,* and *r* represent the principal, time, and rate of interest respectively.

It is given that the principal increases continuously at the rate of *r*% per year.

Integrating both sides, we get:

It is given that when *t* = 0, *p* = 100.

â‡’ 100 = *e ^{k}* â€¦ (2)

Now, if *t* = 10, then *p* = 2 Ã— 100 = 200.

Therefore, equation (1) becomes:

Hence, the value of *r* is 6.93%.

**Question 56: In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years**.

**ANSWER : -** Let *p* and *t* be the principal and time respectively.

It is given that the principal increases continuously at the rate of 5% per year.

Integrating both sides, we get:

Now, when *t* = 0, *p* = 1000.

â‡’ 1000 = *e*^{C} â€¦ (2)

At *t* = 10, equation (1) becomes:

Hence, after 10 years the amount will worth Rs 1648.

**Question 57: In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?**

**ANSWER : -** Let *y* be the number of bacteria at any instant *t*.

It is given that the rate of growth of the bacteria is proportional to the number present.

Integrating both sides, we get:

Let *y*_{0} be the number of bacteria at *t* = 0.

â‡’ log *y*_{0} = C

Substituting the value of C in equation (1), we get:

Also, it is given that the number of bacteria increases by 10% in 2 hours.

Substituting this value in equation (2), we get:

Therefore, equation (2) becomes:

Now, let the time when the number of bacteria increases from 100000 to 200000 be *t*_{1}.

â‡’ *y* = 2*y*_{0} at *t* = *t*_{1}

From equation (4), we get:

Hence, in hours the number of bacteria increases from 100000 to 200000.

**Question 58: The general solution of the differential equation **

**A.**

**B.**

**C.**

**D. **

**ANSWER : -**

Integrating both sides, we get:

Hence, the correct answer is A.

**Question 59: **

**ANSWER : -** The given differential equation i.e., (*x*^{2} *xy*) *dy* = (*x*^{2} *y*^{2}) *dx* can be written as:

This shows that equation (1) is a homogeneous equation.

To solve it, we make the substitution as: *y *= *vx*

Differentiating both sides with respect to *x*, we get:

Substituting the values of *v* and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

**Question 60: **

**ANSWER : -** The given differential equation is:

Thus, the given equation is a homogeneous equation.

To solve it, we make the substitution as: *y *= *vx*

Differentiating both sides with respect to *x*, we get:

Substituting the values of *y* and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

**Question 61: **

**ANSWER : -** The given differential equation is:

Thus, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y *= *vx*

Substituting the values of *y* and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

**Question 62: **

**ANSWER : -** The given differential equation is:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as: *y *= *vx*

Substituting the values of *y* and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

**Question 63: **

**ANSWER : -** The given differential equation is:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as: *y *= *vx*

Substituting the values of *y* and in equation (1), we get:

Integrating both sides, we get:

This is the required solution for the given differential equation.

**Question 64: **

**ANSWER : -**

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y *= *vx*

Substituting the values of *v *and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

**Question 65: **

**ANSWER : -** The given differential equation is:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y *= *vx*

Substituting the values of *y* and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

**Question 66: **

**ANSWER : -**

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y *= *vx*

Substituting the values of *y* and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

**Question 67: **

**ANSWER : -**

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as: *y *= *vx*

Substituting the values of *y* and in equation (1), we get:

Integrating both sides, we get:

Therefore, equation (1) becomes:

This is the required solution of the given differential equation.

**Question 68: **

**ANSWER : -**

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*x *= *vy*

Substituting the values of *x* and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

**Question 69: **

**ANSWER : -**

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y *= *vx*

Substituting the values of *y* and in equation (1), we get:

Integrating both sides, we get:

Now, *y* = 1 at *x* = 1.

Substituting the value of 2*k* in equation (2), we get:

This is the required solution of the given differential equation.

**Question 70: **

**ANSWER : -**

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as: *y *= *vx*

Substituting the values of *y* and in equation (1), we get:

Integrating both sides, we get:

Now, *y* = 1 at *x* = 1.

Substituting in equation (2), we get:

This is the required solution of the given differential equation.

**Question 71: **

**ANSWER : -**

Therefore, the given differential equation is a homogeneous equation.

To solve this differential equation, we make the substitution as:

*y *= *vx*

Substituting the values of *y* and in equation (1), we get:

Integrating both sides, we get:

Now, .

Substituting C = *e* in equation (2), we get:

This is the required solution of the given differential equation.

**Question 72: **

**ANSWER : -**

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y *= *vx*

Substituting the values of *y* and in equation (1), we get:

Integrating both sides, we get:

This is the required solution of the given differential equation.

Now, *y* = 0 at *x* = 1.

Substituting C = *e* in equation (2), we get:

This is the required solution of the given differential equation.

**Question 73: **

**ANSWER : -**

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y *= *vx*

Substituting the value of *y* and in equation (1), we get:

Integrating both sides, we get:

Now, *y* = 2 at *x* = 1.

Substituting *C* = â€“1 in equation (2), we get:

This is the required solution of the given differential equation.

**Question 74: A homogeneous differential equation of the form can be solved by making the substitution**

**A. y = vx **

**B. v = yx**

**C. x = vy **

**D. x = v**

**ANSWER : -** For solving the homogeneous equation of the form, we need to make the substitution as *x* = *vy*.Hence, the correct answer is C.

**Question 75: Which of the following is a homogeneous differential equation?**

**A.**

**B.**

**C.**

**D.**

**ANSWER : -** Function F(*x*, *y*) is said to be the homogenous function of degree *n,* if

F(Î»*x*, Î»*y*) = Î»* ^{n}* F(

Consider the equation given in alternativeD:

Hence, the differential equation given in alternative **D** is a homogenous equation.

**Question 76: **

**ANSWER : -** The given differential equation is

This is in the form of

The solution of the given differential equation is given by the relation,

Therefore, equation (1) becomes:

This is the required general solution of the given differential equation.

**Question 77: **

**ANSWER : -** The given differential equation is

The solution of the given differential equation is given by the relation,

This is the required general solution of the given differential equation.

**Question 78: **

**ANSWER : -** The given differential equation is:

The solution of the given differential equation is given by the relation,

This is the required general solution of the given differential equation.

**Question 79: **

**ANSWER : -** The given differential equation is:

The general solution of the given differential equation is given by the relation,

**Question 80: **

**ANSWER : -** The given differential equation is:

This equation is in the form of:

The general solution of the given differential equation is given by the relation,

Therefore, equation (1) becomes:

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