Question 66 of 96intermediate🔧 ApplyLong Answer4 marks

For each of the differential equations given in Exercises 1 to 12, find the general solution:

dydx+yx=x2\frac{dy}{dx} + \frac{y}{x} = x^2
Correct Answer

Step 1: The given differential equation is

dydx+yx=x2.\frac{dy}{dx} + \frac{y}{x} = x^2.

This is a linear differential equation of the form

dydx+P(x)y=Q(x),\frac{dy}{dx} + P(x)y = Q(x),

where P(x)=1xP(x) = \frac{1}{x} and Q(x)=x2Q(x) = x^2.

Step 2: Find the integrating factor (IF). The IF is given by e∫P(x) dxe^{\int P(x)\,dx}. Here,

∫P(x) dx=∫1x dx=ln⁡∣x∣.\int P(x)\,dx = \int \frac{1}{x}\,dx = \ln|x|.

Thus,

IF=eln⁡∣x∣=∣x∣.\text{IF} = e^{\ln|x|} = |x|.

For simplicity, we take x>0x > 0 and use IF=x\text{IF} = x.

Step 3: Multiply both sides of the differential equation by the IF xx:

xdydx+y=x3.x \frac{dy}{dx} + y = x^3.

Step 4: Observe that the left-hand side is the derivative of the product xyx y with respect to xx, i.e.,

ddx(xy)=xdydx+y.\frac{d}{dx}(x y) = x \frac{dy}{dx} + y.

Hence, the equation becomes:

ddx(xy)=x3.\frac{d}{dx}(x y) = x^3.

Step 5: Integrate both sides with respect to xx:

xy=∫x3 dx=x44+C,x y = \int x^3\,dx = \frac{x^4}{4} + C,

where CC is the constant of integration.

Step 6: Solve for yy by dividing both sides by xx (assuming x≠0x \neq 0):

y=x34+Cx.y = \frac{x^3}{4} + \frac{C}{x}.

Thus, the general solution of the given differential equation is

y=x34+Cx.y = \frac{x^3}{4} + \frac{C}{x}.
Exercise: EXERCISE 9.5 | Q: 3 | (Page 29)
For More Understanding

Explanation

This question is from Exercise 9.5 of the NCERT textbook. The differential equation is linear, and the standard method involves finding an integrating factor.

The steps shown follow the typical procedure: identify PP and QQ, compute IF=e∫P dx\text{IF} = e^{\int P\,dx}, multiply through, recognize the derivative of the product, integrate, and solve for yy.

The final answer matches the expected general solution.

Solution Steps

  1. Step 1: Identify the equation as linear:

    dydx+1xy=x2.\frac{dy}{dx} + \frac{1}{x}y = x^2.

  2. Step 2: Compute integrating factor:

    IF=e∫1x dx=eln⁡x=x.\text{IF} = e^{\int \frac{1}{x}\,dx} = e^{\ln x} = x.

  3. Step 3: Multiply both sides by IF:

    xdydx+y=x3.x \frac{dy}{dx} + y = x^3.

  4. Step 4: Recognize left side as ddx(xy)\frac{d}{dx}(x y).

  5. Step 5: Integrate:

    xy=∫x3 dx=x44+C.x y = \int x^3\,dx = \frac{x^4}{4} + C.

  6. Step 6: Solve for yy:

    y=x34+Cx.y = \frac{x^3}{4} + \frac{C}{x}.
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